Determining Limits Graphically

A limit describes what $f(x)$ gets close to as $x$ gets close to some number $a$:

$$\lim_{x \to a} f(x) = L$$

reads "the limit of $f(x)$ as $x$ approaches $a$ is $L$."

The key idea: the limit doesn't care what happens at $a$, only near $a$. The function may not even be defined there.

Reading a limit from a graph

Put a finger on the graph on each side of $x = a$ and slide toward $a$. The height your fingers approach is the limit.

Here, as $x \to 2$ from either side, the line heads to height $3$. So $\lim_{x \to 2} f(x) = 3$, even though the open circle shows the point $(2, 3)$ is missing and the filled dot says $f(2) = 1$. The limit and the value are different things.

One-sided limits

• $\lim_{x \to a^-} f(x)$: approach from the left (values less than $a$).
• $\lim_{x \to a^+} f(x)$: approach from the right.

From the left, $g$ approaches $1$; from the right, it approaches $3$:

$$\lim_{x \to 1^-} g(x) = 1 \qquad \lim_{x \to 1^+} g(x) = 3$$

The two-sided limit exists only when both one-sided limits exist and are equal, so $\lim_{x \to 1} g(x)$ does not exist.

Other ways a limit fails

• Unbounded: the graph shoots up or down without end near $a$, as with a vertical asymptote. We write $\lim = \infty$ to describe how it fails (section 04.05).
• Oscillation: $\sin\frac{1}{x}$ wiggles faster and faster near $0$ and never settles.

Tables

Without a graph, compute values close to $a$ on both sides:

$x$	$2.9$	$2.99$	$3.01$	$3.1$
$f(x)$	$5.8$	$5.98$	$6.02$	$6.2$

The outputs close in on $6$. Tables and graphs only suggest a limit; the next sections show how to compute one exactly.