Unit 1 Beta

Limits and Continuity: the one-page sheet

1.1 Can Change Happen at an Instant?

An average rate of change, (f(b) − f(a))/(b − a) , is a slope over an interval, with units of output per input. A rate at one instant cannot be a quotient, since nothing changes in zero time; instead, the averages over shrinking intervals close in on one number, the instantaneous rate of change. The next topic gives that closing in a name and a notation.

The average rate of change of $f$ over $[a, b]$ is $\frac{f(b) - f(a)}{b - a}$ : the slope of the line through $(a, f(a))$ and $(b, f(b))$ . Its units are output units per input unit.
A rate of change at one instant cannot be computed as a quotient, because the interval would have length 0.
The instantaneous rate of change at $t = a$ is the number the average rates over shorter and shorter intervals around $a$ close in on.
From a table, estimate the rate at an instant with the shortest interval that contains it; from both sides, average the two closest rates.
A negative rate means the quantity is decreasing; its size tells how fast.

the average rate of change (f(b) − f(a))/(b − a) , the slope of the line through two points of the graph the quotient becomes zero divided by zero, which has no value the average rates close in on one number it is the instantaneous rate of change: the speed the speedometer shows

average rate of change: The change in a quantity divided by the change in the input over an interval: <math class="calc-m"><semantics><mrow><mfrac><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mo>−</mo><mi>f</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo></mrow><mrow><mi>b</mi><mo>−</mo><mi>a</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{f(b) - f(a)}{b - a}</annotation></semantics></math>. It is the slope of the line through <math class="calc-m"><semantics><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>f</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(a, f(a))</annotation></semantics></math> and <math class="calc-m"><semantics><mrow><mo stretchy="false">(</mo><mi>b</mi><mo>,</mo><mi>f</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(b, f(b))</annotation></semantics></math>, in output units per input unit.
instantaneous rate of change: How fast a quantity is changing at one instant: the number that average rates of change over shorter and shorter intervals around that instant close in on.

1.2 What a Limit Is and How to Write It

lim as x → a, f(x) = L says that f(x) closes in on L as x approaches a from both sides. The value at a never enters: the limit can equal f(a) , differ from it, or exist where f(a) is undefined. Limits can be found from graphs, tables and formulas, and the rest of Unit 1 shows how.

$\lim_{x \to a} f(x) = L$ means: as $x$ gets close to $a$ (from both sides, $x \ne a$ ), $f(x)$ gets close to $L$ , as close as we like.
The limit at $a$ and the value $f(a)$ are different questions. A hole can have a limit and no value; a filled dot off the curve gives a value that is not the limit.
Read the notation from the bottom up: the input heads to $a$ , the output heads to $L$ .
Write "lim" with its arrow on every line until the line where you substitute and get a number; then drop it.

watch where the outputs f(x) go we write lim as x → a, f(x) = L : the limit of f(x) as x approaches a is L the limit can differ from f(a) , or exist when f(a) is not defined the next four topics teach each way

limit: <math class="calc-m"><semantics><mrow><munder><mo movablelimits="false">lim</mo><mrow><mi>x</mi><mo>→</mo><mi>a</mi></mrow></munder><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>L</mi></mrow><annotation encoding="application/x-tex">\lim_{x \to a} f(x) = L</annotation></semantics></math> means the values <math class="calc-m"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x)</annotation></semantics></math> get as close to <math class="calc-m"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math> as we like when <math class="calc-m"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math> is close enough to <math class="calc-m"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math> (but not equal to <math class="calc-m"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math>). The value <math class="calc-m"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(a)</annotation></semantics></math> plays no part.

1.3 Reading Limits from a Graph

Read a graph's limit one side at a time. The left-hand and right-hand limits are the heights the graph heads to from each side. If they exist and agree, that is the limit; if they differ, or the outputs are unbounded or oscillate, the limit does not exist. Dots give the value f(a) , which never affects the limit.

$\lim_{x \to a^-} f(x)$ uses only $x < a$ ; $\lim_{x \to a^+} f(x)$ uses only $x > a$ . Follow the graph's heights on that side to the edge at $x = a$ .
$\lim_{x \to a} f(x) = L$ exactly when both one-sided limits equal $L$ .
A limit fails to exist when the one-sided limits differ (a jump), when the outputs grow without bound, or when they oscillate.
Filled and open dots tell you $f(a)$ ; they never decide the limit.
A justification names both one-sided limits: "From the left $f(x) \to 3$ ; from the right $f(x) \to 1$ ; since $3 \ne 1$ , the limit does not exist."

the heights give the left-hand limit, lim as x → a⁻, f(x) the heights give the right-hand limit, lim as x → a⁺, f(x) the two-sided limit exists and equals that number the limit does not exist

one-sided limit: The limit as <math class="calc-m"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math> approaches <math class="calc-m"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math> from one side only: <math class="calc-m"><semantics><mrow><munder><mo movablelimits="false">lim</mo><mrow><mi>x</mi><mo>→</mo><msup><mi>a</mi><mo>−</mo></msup></mrow></munder><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\lim_{x \to a^-} f(x)</annotation></semantics></math> uses <math class="calc-m"><semantics><mrow><mi>x</mi><mo><</mo><mi>a</mi></mrow><annotation encoding="application/x-tex">x < a</annotation></semantics></math> (left-hand limit) and <math class="calc-m"><semantics><mrow><munder><mo movablelimits="false">lim</mo><mrow><mi>x</mi><mo>→</mo><msup><mi>a</mi><mo>+</mo></msup></mrow></munder><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\lim_{x \to a^+} f(x)</annotation></semantics></math> uses <math class="calc-m"><semantics><mrow><mi>x</mi><mo>></mo><mi>a</mi></mrow><annotation encoding="application/x-tex">x > a</annotation></semantics></math> (right-hand limit).
limit does not exist: A limit does not exist (DNE) when the outputs do not close in on one number: the one-sided limits differ, the outputs grow without bound, or they oscillate.
unbounded behavior: Outputs that grow larger and larger in size without any bound, as near a vertical asymptote. Oscillating behavior is outputs that keep swinging between values and never settle.

1.4 Estimating Limits from Tables

A table estimates a limit when it lists outputs for inputs closing in on a from both sides. If both sides settle on the same number, that is the estimate; if they differ, grow without bound or never settle, the limit appears not to exist. A table suggests; it does not prove, so check closer inputs or use algebra when in doubt.

To estimate $\lim_{x \to a} f(x)$ , evaluate $f$ at inputs like $a - 0.1, a - 0.01, a - 0.001$ and $a + 0.001, a + 0.01, a + 0.1$ .
Read each side separately, exactly as on a graph: the left-hand limit uses inputs below $a$ , the right-hand limit inputs above $a$ .
The estimate is the number the outputs approach, not the last value in the table. A value $f(a)$ in the table never decides the limit.
A table can mislead if it stops too early. If the values have not settled, test inputs even closer to $a$ .

the outputs show each one-sided trend that number is a numerical estimate of the limit the table suggests the limit does not exist it suggests a limit but cannot prove it; confirm with a graph or algebra

numerical estimate: An estimate of a limit from a table of values: the number the outputs approach as the inputs get closer and closer to <math class="calc-m"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math> from both sides.

1.5 Limit Laws

The limit laws let you split a limit over sums, differences, constant multiples, products, quotients (with a nonzero denominator limit), powers, roots and compositions. Together they justify direct substitution for polynomials and for rational functions where the denominator is not 0. When a denominator heads to 0, look closer: a nonzero numerator means no finite limit.

Sum, difference, constant multiple, product: if both limits exist, combine them the same way. $\lim (f + g)$ $= \lim f + \lim g$ .
Quotient: $\lim \frac{f}{g}$ $= \frac{\lim f}{\lim g}$ only when $\lim g \ne 0$ .
Powers and roots: $\lim [f(x)]^n = L^n$ , and $\lim \sqrt{f(x)} = \sqrt{L}$ when $L > 0$ .
Direct substitution: for a polynomial, or a rational function with a nonzero denominator at $a$ , the limit is the value at $a$ .
For a composite, find the inside limit first: if $g(x) \to b$ and $f$ is continuous at $b$ , then $f(g(x)) \to f(b)$ .
For a piecewise function, use each side's formula for its one-sided limit.

the limit of f ± g , fg and kf is L ± M , LM and kL the limit of f/g is L/M their limits come from substitution, as long as the denominator is not 0 at a the quotient law is silent: a nonzero top means no finite limit; a top also heading to 0 needs the algebra of Topic 1.6

limit laws: Rules that combine limits that exist: the limit of a sum, difference, constant multiple, product or power is the same combination of the limits; a quotient's limit is the quotient of the limits when the denominator's limit is not 0.
direct substitution: Finding <math class="calc-m"><semantics><mrow><munder><mo movablelimits="false">lim</mo><mrow><mi>x</mi><mo>→</mo><mi>a</mi></mrow></munder><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\lim_{x \to a} f(x)</annotation></semantics></math> by evaluating <math class="calc-m"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(a)</annotation></semantics></math>. It is justified for polynomials, and for rational, root and similar functions at points where the formula is defined (no zero denominator).

1.6 Limits by Algebra: Factor, Rationalize, Simplify

When direct substitution gives the indeterminate form 0/0 , rewrite: factor and cancel, multiply by a conjugate, or simplify a complex fraction. The simplified expression matches the original except at x = a , so it has the same limit, which substitution now finds. Write every step with "lim" and true equals signs.

$\frac{0}{0}$ is an indeterminate form: it means "rewrite first", never 0, 1 or "does not exist".
Canceling is legal in a limit because the limit never uses $x = a$ , and for $x \ne a$ the canceled factor is not 0.
Toolkit: factor and cancel; multiply by a conjugate to clear a square root; combine a complex fraction; expand powers completely.
After simplifying, a nonzero number over 0 means no finite limit (unbounded); check one-sided signs if asked.
Keep " $\lim_{x \to a}$ " on every line until you substitute; never write $\frac{0}{0} = \ldots$ .

an indeterminate form: the limit is not yet known they share a factor x − a (for polynomials), possibly hidden behind a square root or a fraction the new expression equals the old one for every x ≠ a , so it has the same limit the limit appears (or a nonzero over zero shows there is no finite limit)

indeterminate form: A result like <math class="calc-m"><semantics><mrow><mfrac><mn>0</mn><mn>0</mn></mfrac></mrow><annotation encoding="application/x-tex">\frac{0}{0}</annotation></semantics></math> from substituting into a limit. It does not determine the limit; the expression must be rewritten (factored, rationalized, simplified) first.

1.7 Choosing a Way to Find a Limit

Substitute first and read the result. A number is the limit. 0/0 means rewrite: factor and cancel, multiply by a conjugate, or combine fractions, then substitute again. A nonzero number over 0 means the function is unbounded there. At a break point, compare the one-sided limits.

Substitute first. It is the fastest test, and its result tells you what to do next.
$\frac{0}{0}$ is an indeterminate form: not an answer, a signal to rewrite.
A nonzero number over 0 means the function is unbounded near $c$ ; no algebra will make the limit a number.
At a break point, the two sides can follow different rules. Find both one-sided limits.
Write the limit symbol on every line until you substitute: the simplified expression equals the original only for $x \ne c$ .

that number is the limit, as long as the expression is defined near c the numerator and denominator share a hidden zero factor, so rewrite (factor, conjugate, common denominator) and substitute again the denominator shrinks while the numerator does not, so the function is unbounded near c and the limit does not exist find the left-hand and right-hand limits separately; equal means the limit exists

limit strategy: A plan for finding a limit: substitute first, then let the result decide the next step. A number is the limit; <math class="calc-m"><semantics><mrow><mfrac><mn>0</mn><mn>0</mn></mfrac></mrow><annotation encoding="application/x-tex">\frac{0}{0}</annotation></semantics></math> means rewrite (factor, use a conjugate, combine fractions) and substitute again; a nonzero number over 0 means the function is unbounded there; a break point means compare the one-sided limits.

1.8 The Squeeze Theorem

If g(x) ≤ f(x) ≤ h(x) near c and g and h have the same limit L at c , then f has limit L too. Bounds usually come from −1 ≤ sin ≤ 1 . The squeeze gives lim as x → 0, (sin x)/x = 1 in radians, and (sin (kx))/x → k .

Squeeze theorem: a function trapped between two functions with the same limit $L$ at $c$ also has limit $L$ at $c$ .
Both bounds must approach the same value. Bounds with different limits give no conclusion.
Build bounds from $-1 \le \sin\theta \le 1$ and $-1 \le \cos\theta \le 1$ , multiplied by a factor that is not negative (use $|x|$ , not $x$ ).
The special limit $\lim_{x \to 0} \frac{\sin x}{x}$ $= 1$ comes from the squeeze. Match the inside: $\frac{\sin(kx)}{x}$ $= k \cdot \frac{\sin(kx)}{kx} \to k$ .
A justification states the inequality, shows the two bounds have the same limit, and names the squeeze theorem.

every value of f near c lies between the values of g and h the gap between them shrinks to nothing near c lim as x → c, f(x) = L : the squeeze theorem lim as x → 0, (sin x)/x = 1 (with x in radians)

squeeze theorem: If <math class="calc-m"><semantics><mrow><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>≤</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>≤</mo><mi>h</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">g(x) \le f(x) \le h(x)</annotation></semantics></math> for every <math class="calc-m"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math> near <math class="calc-m"><semantics><mrow><mi>c</mi></mrow><annotation encoding="application/x-tex">c</annotation></semantics></math> (except possibly at <math class="calc-m"><semantics><mrow><mi>c</mi></mrow><annotation encoding="application/x-tex">c</annotation></semantics></math>), and <math class="calc-m"><semantics><mrow><munder><mo movablelimits="false">lim</mo><mrow><mi>x</mi><mo>→</mo><mi>c</mi></mrow></munder><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\lim_{x \to c} g(x)</annotation></semantics></math><math class="calc-m"><semantics><mrow><mo form="infix">=</mo><munder><mo movablelimits="false">lim</mo><mrow><mi>x</mi><mo>→</mo><mi>c</mi></mrow></munder><mi>h</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">= \lim_{x \to c} h(x)</annotation></semantics></math><math class="calc-m"><semantics><mrow><mo form="infix">=</mo><mi>L</mi></mrow><annotation encoding="application/x-tex">= L</annotation></semantics></math>, then <math class="calc-m"><semantics><mrow><munder><mo movablelimits="false">lim</mo><mrow><mi>x</mi><mo>→</mo><mi>c</mi></mrow></munder><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>L</mi></mrow><annotation encoding="application/x-tex">\lim_{x \to c} f(x) = L</annotation></semantics></math>. Also called the sandwich theorem.

1.9 Limits Across Graphs, Tables and Formulas

A limit can be read from a graph (the height the curve approaches), estimated from a table (values on both sides), found exactly from a formula (algebra), and stated in words or notation. Move between them: estimate with a table, confirm with a formula, and check the sides when one function sits inside another.

The same limit can be read from a graph, a table, a formula or words. Each answer must agree.
On a graph, the limit is the height the curve approaches, not the closed dot at the point.
A table suggests a limit; a formula confirms it. Cleverly chosen inputs can make a table mislead, as with $\sin\left(\frac{\pi}{x}\right)$ .
Translate words exactly: "from values less than 4" is $x \to 4^-$ ; the number the outputs approach goes after the equals sign.
For $f(g(x))$ , find the inner limit first and note the side it comes from.

follow the curve toward x = c from each side and read the height it approaches; ignore any dot exactly at c the values suggest the limit, but a table cannot prove it algebra (substitute, factor, rationalize) gives the exact limit and confirms the estimate find where g(x) goes and from which side, then read f near that input

multiple representations: The ways one function can be given: a graph, a table of values, a formula, or words. A limit can be read or estimated from each one; an exam question often gives one representation and asks about another.

1.10 Types of Discontinuity

A discontinuity is a point where the graph breaks. Removable: the limit exists but the value is missing or different (a hole). Jump: finite one-sided limits that differ. Infinite: unbounded near the point (a vertical asymptote). Classify by finding the one-sided limits and comparing them with f(c) .

A function is continuous at $x = c$ when $\lim_{x \to c} f(x)$ exists and equals $f(c)$ ; its graph has no break there. A discontinuity is a point where this fails.
Removable: the limit exists, but the point is missing or misplaced (a hole).
Jump: both one-sided limits are finite and different.
Infinite: the function is unbounded near $c$ (a vertical asymptote).
To classify, find the one-sided limits first, then compare them with $f(c)$ .

the graph has a hole: a removable discontinuity the graph steps from one height to another: a jump discontinuity the graph has a vertical asymptote x = c : an infinite discontinuity expect a hole; a factor that stays in the denominator gives an asymptote

discontinuity: A point where a function is not continuous: its graph breaks there. At <math class="calc-m"><semantics><mrow><mi>x</mi><mo>=</mo><mi>c</mi></mrow><annotation encoding="application/x-tex">x = c</annotation></semantics></math> this happens when <math class="calc-m"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(c)</annotation></semantics></math> is undefined, when <math class="calc-m"><semantics><mrow><munder><mo movablelimits="false">lim</mo><mrow><mi>x</mi><mo>→</mo><mi>c</mi></mrow></munder><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\lim_{x \to c} f(x)</annotation></semantics></math> does not exist, or when the two are not equal.
removable discontinuity: A discontinuity at <math class="calc-m"><semantics><mrow><mi>x</mi><mo>=</mo><mi>c</mi></mrow><annotation encoding="application/x-tex">x = c</annotation></semantics></math> where <math class="calc-m"><semantics><mrow><munder><mo movablelimits="false">lim</mo><mrow><mi>x</mi><mo>→</mo><mi>c</mi></mrow></munder><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\lim_{x \to c} f(x)</annotation></semantics></math> exists, but <math class="calc-m"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(c)</annotation></semantics></math> is undefined or not equal to it. The graph has a hole, possibly with a dot elsewhere above or below it.
jump discontinuity: A discontinuity at <math class="calc-m"><semantics><mrow><mi>x</mi><mo>=</mo><mi>c</mi></mrow><annotation encoding="application/x-tex">x = c</annotation></semantics></math> where both one-sided limits exist as finite numbers but are not equal, so the graph jumps from one height to another.
infinite discontinuity: A discontinuity at <math class="calc-m"><semantics><mrow><mi>x</mi><mo>=</mo><mi>c</mi></mrow><annotation encoding="application/x-tex">x = c</annotation></semantics></math> where the function is unbounded on at least one side of <math class="calc-m"><semantics><mrow><mi>c</mi></mrow><annotation encoding="application/x-tex">c</annotation></semantics></math>. The graph has a vertical asymptote <math class="calc-m"><semantics><mrow><mi>x</mi><mo>=</mo><mi>c</mi></mrow><annotation encoding="application/x-tex">x = c</annotation></semantics></math>.

1.11 Continuity at a Point

f is continuous at c when f(c) is defined, lim as x → c, f(x) exists, and the two are equal. Check the three conditions in order and state each one with its number. Continuity at c is exactly what makes lim as x → c, f(x) = f(c) .

Continuous at $c$ means: (1) $f(c)$ is defined, (2) $\lim_{x \to c} f(x)$ exists, (3) $\lim_{x \to c} f(x)$ $= f(c)$ .
Check the conditions in order. The first one that fails tells you the reason, and the kind of discontinuity.
At the break point of a piecewise function, find both one-sided limits from the pieces, then compare with the value the function is given there.
A justification shows all three conditions with numbers: " $f(2) = 3$ , $\lim_{x \to 2} f(x) = 3$ , so $\lim_{x \to 2} f(x)$ $= f(2)$ ."
If $f$ is continuous at $c$ , its limit at $c$ is just $f(c)$ . That is why direct substitution works.

condition 1 fails: there is no point on the graph at x = c condition 2 fails: lim as x → c, f(x) does not exist condition 3 fails: the point is not where the curve arrives f is continuous at c , and lim as x → c, f(x) = f(c) , so the limit can be found by substitution

definition of continuity: A function <math class="calc-m"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> is continuous at <math class="calc-m"><semantics><mrow><mi>x</mi><mo>=</mo><mi>c</mi></mrow><annotation encoding="application/x-tex">x = c</annotation></semantics></math> when three conditions hold: <math class="calc-m"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(c)</annotation></semantics></math> is defined, <math class="calc-m"><semantics><mrow><munder><mo movablelimits="false">lim</mo><mrow><mi>x</mi><mo>→</mo><mi>c</mi></mrow></munder><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\lim_{x \to c} f(x)</annotation></semantics></math> exists, and <math class="calc-m"><semantics><mrow><munder><mo movablelimits="false">lim</mo><mrow><mi>x</mi><mo>→</mo><mi>c</mi></mrow></munder><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\lim_{x \to c} f(x)</annotation></semantics></math><math class="calc-m"><semantics><mrow><mo form="infix">=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">= f(c)</annotation></semantics></math>.

1.12 Continuity on an Interval

A function is continuous on an interval when it is continuous at every point of it, with one-sided continuity at closed endpoints. The familiar function families are continuous on their domains, and sums, products, quotients and compositions keep that. So you only need to check where a function is undefined and where a piecewise rule switches.

Continuous on an interval: continuous at every point of it. At a closed endpoint, only the one-sided limit from inside must match the value.
Polynomials, $e^x$ , $\sin x$ and $\cos x$ are continuous on $(-\infty, \infty)$ .
Rational functions, roots, $\ln x$ and $\tan x$ are continuous on their domains: avoid zeros of denominators, negative inputs to even roots, inputs to $\ln$ that are not positive, and zeros of $\cos x$ for $\tan x$ .
Sums, products, quotients (denominator not 0) and compositions of continuous functions are continuous.
For a piecewise function, each piece is handled by the rules above; check each break point with the three conditions.

it is continuous on that interval: its graph has no break there at the endpoints only the one-sided limit from inside counts: lim as x → a⁺, f(x) = f(a) and lim as x → b⁻, f(x) = f(b) are continuous on their domains, so you only check where they are undefined the result is continuous wherever it is defined; only break points of piecewise rules need a separate check

continuous on an interval: A function is continuous on an open interval when it is continuous at every point of it. On a closed interval <math class="calc-m"><semantics><mrow><mo stretchy="false">[</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[a, b]</annotation></semantics></math> it must also be continuous from the inside at each endpoint: <math class="calc-m"><semantics><mrow><munder><mo movablelimits="false">lim</mo><mrow><mi>x</mi><mo>→</mo><msup><mi>a</mi><mo>+</mo></msup></mrow></munder><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\lim_{x \to a^+} f(x)</annotation></semantics></math><math class="calc-m"><semantics><mrow><mo form="infix">=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">= f(a)</annotation></semantics></math> and <math class="calc-m"><semantics><mrow><munder><mo movablelimits="false">lim</mo><mrow><mi>x</mi><mo>→</mo><msup><mi>b</mi><mo>−</mo></msup></mrow></munder><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\lim_{x \to b^-} f(x)</annotation></semantics></math><math class="calc-m"><semantics><mrow><mo form="infix">=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">= f(b)</annotation></semantics></math>.

1.13 Removing a Discontinuity

A removable discontinuity is removed by defining f(c) to be the limit at c . A piecewise function is made continuous at a break point by choosing constants so the two one-sided limits and the value agree; each break point gives one equation. Jumps and infinite discontinuities cannot be removed by choosing a value.

To remove a removable discontinuity, find $L = \lim_{x \to c} f(x)$ and define $f(c) = L$ .
At a break point of a piecewise function, set the left-hand limit, the right-hand limit and the value equal. Each break point gives one equation.
A constant can only move its own piece. If the value at the point and the other piece already disagree, no constant fixes it.
A jump or an infinite discontinuity cannot be removed: the limit does not exist, so no value can equal it.
Write the result as a check: substitute your constant back and show both sides give the same number.

defining (or redefining) f(c) = L makes f continuous at c continuity at the break point gives an equation: the left-hand limit equals the right-hand limit equals the value each break point gives one equation; solve them together no single value can fix it, because the two-sided limit does not exist

removing a discontinuity: Making a function continuous at a point by choosing its value there, or a constant in its formula. A removable discontinuity is removed by defining <math class="calc-m"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(c)</annotation></semantics></math><math class="calc-m"><semantics><mrow><mo form="infix">=</mo><munder><mo movablelimits="false">lim</mo><mrow><mi>x</mi><mo>→</mo><mi>c</mi></mrow></munder><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">= \lim_{x \to c} f(x)</annotation></semantics></math>; a piecewise function is made continuous at a break point by choosing constants so both pieces and the value agree. Jump and infinite discontinuities cannot be removed this way.

1.14 Infinite Limits and Vertical Asymptotes

An infinite limit, lim as x → c, f(x) = ±∞ , says f grows without bound near c ; the limit does not exist as a number. Find each side's sign from the signs of the numerator and the shrinking denominator. Any infinite one-sided limit at c makes x = c a vertical asymptote; a canceled factor makes a hole instead.

$\lim_{x \to c} f(x) = \infty$ means $f(x)$ grows beyond every bound as $x$ nears $c$ . It is an infinite limit: the limit does not exist, and this says how.
Find each side by signs: a nonzero numerator over a denominator approaching 0 from the positive or negative side.
If the two sides agree, write $\lim_{x \to c} f(x) = \infty$ (or $-\infty$ ). If they disagree, give each one-sided limit.
$x = c$ is a vertical asymptote when at least one one-sided limit at $c$ is $\infty$ or $-\infty$ .
For a rational function, cancel common factors first: a canceled factor gives a hole, a factor left in the denominator gives an asymptote.

the quotient grows without bound in size each one-sided limit is ∞ or −∞ the line x = c is a vertical asymptote lim as x → c, f(x) = ∞ describes how the limit fails to exist; never write f(c) = ∞

infinite limit: A limit in which the values of <math class="calc-m"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x)</annotation></semantics></math> grow without bound as <math class="calc-m"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math> approaches <math class="calc-m"><semantics><mrow><mi>c</mi></mrow><annotation encoding="application/x-tex">c</annotation></semantics></math>: <math class="calc-m"><semantics><mrow><munder><mo movablelimits="false">lim</mo><mrow><mi>x</mi><mo>→</mo><mi>c</mi></mrow></munder><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi mathvariant="normal">∞</mi></mrow><annotation encoding="application/x-tex">\lim_{x \to c} f(x) = \infty</annotation></semantics></math> (or <math class="calc-m"><semantics><mrow><mo form="prefix" lspace="0" rspace="0">−</mo><mi mathvariant="normal">∞</mi></mrow><annotation encoding="application/x-tex">-\infty</annotation></semantics></math>). It is a way of describing a limit that does not exist; it gives the graph a vertical asymptote <math class="calc-m"><semantics><mrow><mi>x</mi><mo>=</mo><mi>c</mi></mrow><annotation encoding="application/x-tex">x = c</annotation></semantics></math>.

1.15 Limits at Infinity and Horizontal Asymptotes

A limit at infinity describes where f(x) heads as x → ∞ or x → −∞ ; a finite one gives a horizontal asymptote. Divide by the dominant term: for rational functions compare degrees. Watch √x² = |x| on the left. In the long run ln x grows slower than any power and e^x faster than any power.

$\lim_{x \to \infty} f(x) = L$ means $f(x)$ approaches $L$ as $x$ grows without bound. A finite limit at $\infty$ or $-\infty$ gives a horizontal asymptote $y = L$ .
For a rational function, divide by the highest power of $x$ in the denominator. Equal degrees: ratio of leading coefficients. Bottom higher: 0. Top higher: no horizontal asymptote.
$\sqrt{x^2} = |x|$ . As $x \to -\infty$ , $\sqrt{x^2} = -x$ , which can flip the sign of the answer.
Dominant terms: in the long run $\ln x$ loses to every positive power, and $e^x$ beats every power.
Vertical asymptote: $\lim_{x \to c} f(x) = \pm\infty$ . Horizontal asymptote: $\lim_{x \to \pm\infty} f(x) = L$ . Do not mix them up.

terms like 1/x , 5/x² and e^(−x) shrink to 0 every other term becomes one of those shrinking terms, and the limit can be read off y = L is a horizontal asymptote; the graph levels off there (and may cross it in the middle) the faster one dominates: as x → ∞ , ln x loses to any power x^p ( p > 0 ), and x^p loses to e^x

limit at infinity: The value <math class="calc-m"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x)</annotation></semantics></math> approaches as <math class="calc-m"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math> grows without bound: <math class="calc-m"><semantics><mrow><munder><mo movablelimits="false">lim</mo><mrow><mi>x</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></munder><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>L</mi></mrow><annotation encoding="application/x-tex">\lim_{x \to \infty} f(x) = L</annotation></semantics></math> (or as <math class="calc-m"><semantics><mrow><mi>x</mi><mo>→</mo><mo form="prefix" lspace="0" rspace="0">−</mo><mi mathvariant="normal">∞</mi></mrow><annotation encoding="application/x-tex">x \to -\infty</annotation></semantics></math>). A finite limit at infinity gives the horizontal asymptote <math class="calc-m"><semantics><mrow><mi>y</mi><mo>=</mo><mi>L</mi></mrow><annotation encoding="application/x-tex">y = L</annotation></semantics></math>; these limits describe the end behavior of a function.
dominant term: The term that grows fastest as <math class="calc-m"><semantics><mrow><mi>x</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow><annotation encoding="application/x-tex">x \to \infty</annotation></semantics></math> and so decides a limit at infinity. In the long run <math class="calc-m"><semantics><mrow><mi>ln</mi><mo lspace="0" rspace="0.1667em">⁡</mo><mi>x</mi></mrow><annotation encoding="application/x-tex">\ln x</annotation></semantics></math> grows slower than any positive power of <math class="calc-m"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math>, a higher power beats a lower one, and <math class="calc-m"><semantics><mrow><msup><mi>e</mi><mi>x</mi></msup></mrow><annotation encoding="application/x-tex">e^x</annotation></semantics></math> beats every power.

1.16 The Intermediate Value Theorem

If f is continuous on [a, b] and d is between f(a) and f(b) , then f(c) = d for at least one c in (a, b) . A justification states both hypotheses with reasons and numbers, names the theorem, and concludes "for some c in (a, b) ". Without continuity on the closed interval, the conclusion can fail.

Intermediate Value Theorem (IVT): if $f$ is continuous on $[a, b]$ and $d$ is between $f(a)$ and $f(b)$ , then $f(c) = d$ for at least one $c$ in $(a, b)$ .
Hypotheses come first: continuity on the closed interval, with a reason ("polynomial", "stated in the problem"), and $d$ between the endpoint values, with the numbers.
The conclusion is existence: at least one $c$ . The theorem does not say where, or how many.
With a table, apply the theorem on each pair of neighboring inputs; non-overlapping intervals give separate solutions.
Finding $c$ is a separate job: algebra, or a calculator (three decimal places).

its graph has no break from (a, f(a)) to (b, f(b)) the horizontal line y = d separates the two endpoints of the graph it must cross the line: f(c) = d for at least one c in (a, b) the theorem promises nothing; the value may or may not be reached

hypothesis: A condition a theorem needs before its conclusion can be used. For the Intermediate Value Theorem the hypotheses are that <math class="calc-m"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> is continuous on the closed interval <math class="calc-m"><semantics><mrow><mo stretchy="false">[</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[a, b]</annotation></semantics></math> and that <math class="calc-m"><semantics><mrow><mi>d</mi></mrow><annotation encoding="application/x-tex">d</annotation></semantics></math> is between <math class="calc-m"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(a)</annotation></semantics></math> and <math class="calc-m"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(b)</annotation></semantics></math>. If a hypothesis fails, the theorem promises nothing.
Intermediate Value Theorem: The Intermediate Value Theorem: if <math class="calc-m"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> is continuous on <math class="calc-m"><semantics><mrow><mo stretchy="false">[</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[a, b]</annotation></semantics></math> and <math class="calc-m"><semantics><mrow><mi>d</mi></mrow><annotation encoding="application/x-tex">d</annotation></semantics></math> is a number between <math class="calc-m"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(a)</annotation></semantics></math> and <math class="calc-m"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(b)</annotation></semantics></math>, then <math class="calc-m"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo>=</mo><mi>d</mi></mrow><annotation encoding="application/x-tex">f(c) = d</annotation></semantics></math> for at least one <math class="calc-m"><semantics><mrow><mi>c</mi></mrow><annotation encoding="application/x-tex">c</annotation></semantics></math> in <math class="calc-m"><semantics><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(a, b)</annotation></semantics></math>.