What a Limit Is and How to Write It

In the last topic, the average speeds of a falling stone over shorter and shorter intervals closed in on 32 feet per second: 48, 40, 33.6, 32.16, 32.016. We kept saying "closes in on." Calculus gives that phrase a name, the limit, and a notation that you will write thousands of times this year. This page is about what a limit means and how to write it so that it earns credit.

What a limit says

Take a function $f$ and a number $a$. Let $x$ get closer and closer to $a$, from both sides, without ever being equal to $a$. Watch the outputs $f(x)$. If they close in on one number $L$, as close as you like, then $L$ is the limit of $f(x)$ as $x$ approaches $a$. We write $$\lim_{x \to a} f(x) = L$$ and read it as "the limit of $f$ of $x$, as $x$ approaches $a$, equals $L$."

Three parts, each with a job:

• "lim" says we are asking about a trend, not a single value.
• "$x \to a$" underneath says where the input is heading.
• "$= L$" says where the outputs are heading.

The value at the point does not matter

This is the most important fact about limits, and the most surprising. The definition never lets $x$ equal $a$. So the limit cannot depend on $f(a)$.

In the figure, the curve heads to the open circle at $(2, 4)$ from both sides, so $\lim_{x \to 2} g(x) = 4$. The function's actual value at 2 is the filled dot: $g(2) = 1$. Both statements are true at once. Three situations come up again and again:

Why build calculus on an idea that ignores the point itself? Because of Topic 1.1. At the instant $t = 1$, the average-speed formula is zero divided by zero: it has no value at all. The limit is exactly the tool that looks at nearby inputs and never at the problem point.

Three ways to find a limit

A limit can be read from three kinds of information, and the next topics teach each one:

• A graph (Topic 1.3): follow the heights as $x$ slides toward $a$ from the left and from the right.
• A table (Topic 1.4): look at outputs for inputs like 1.9, 1.99, 1.999 and 2.1, 2.01, 2.001.
• A formula (Topics 1.5 and 1.6): use rules and algebra to find the limit exactly.

Sometimes the outputs do not close in on a single number. The two sides might head to different heights, or the outputs might grow without stopping. Then there is no limit; Topic 1.3 shows how to recognize each case.

Reading limits in context

When the function describes something real, a limit statement becomes a sentence about a trend. If $V(t)$ is the volume of water in a tank, in liters, $t$ hours after noon, then $$\lim_{t \to 3} V(t) = 12$$ means: as the time gets closer and closer to 3 p.m., the volume of water gets closer and closer to 12 liters. A careful interpretation names the quantity, its units and the input it approaches. It does not claim that $V(3) = 12$, because the limit statement alone does not say that.

Writing limits the way readers expect

On free-response questions, the notation is part of the answer. Habits that keep your work correct:

• Keep "$\lim_{x \to a}$" in front of the expression on every line until the line where you actually get a number. Then drop it.
• Put parentheses around a sum or difference after "lim": "$\lim_{x \to 2} 5 + x$" is ambiguous, while "$\lim_{x \to 2} (5 + x)$$= 7$" is clear.
• Use the equals sign only between things that are equal. Writing $f(2) = 4$ when you mean $\lim_{x \to 2} f(x) = 4$ is a different, and often false, statement.
• Name the function: "$\lim_{x \to 2} g(x) = 4$", not "the limit is 4."