Math Core

Calculus is built on one question: what value does a function head toward as its input gets closer and closer to some number? That question is called a limit. It lets you talk sensibly about a function near a point where plugging in fails, and it is the key that unlocks slopes of curves, instantaneous speed and much more.

Getting close without arriving

Look at the function

f(x) = \frac{x^2 - 4}{x - 2}.

You can't evaluate $f(2)$ : it gives $\dfrac{0}{0}$ , which is undefined. But nothing stops you from plugging in numbers near $2$ . Here is a table of values from both sides.

$x$	$1.9$	$1.99$	$1.999$	$2$	$2.001$	$2.01$	$2.1$
$f(x)$	$3.9$	$3.99$	$3.999$	undefined	$4.001$	$4.01$	$4.1$

As $x$ closes in on $2$ from the left and from the right, $f(x)$ closes in on $4$ . That's a limit, and you write

\lim_{x \to 2} \frac{x^2 - 4}{x - 2} = 4.

The graph tells the same story. For every $x \ne 2$ , the fraction simplifies to $x + 2$ , so the graph is the line $y = x + 2$ with a single point punched out: a hole at $(2, 4)$ .

y = (x² − 4)/(x − 2) is a line with a hole at (2, 4). The limit as x → 2 is 4.Open in grapher →

Definition

Limit

We write $\displaystyle \lim_{x \to a} f(x) = L$ and say "the limit of $f(x)$ as $x$ approaches $a$ is $L$ " if $f(x)$ can be made as close to $L$ as you like by taking $x$ close enough to $a$ , on both sides, but not equal to $a$ .

The phrase "not equal to $a$ " matters. A limit describes the function's behavior near $a$ . What happens at $a$ (whether $f(a)$ exists, or what its value is) has no effect on the limit.

Limits from graphs

Graphs make the difference between a limit and a function value easy to see. Consider the function $g$ below. It follows the line $y = x + 1$ , except that the point at $x = 1$ has been moved up to $(1, 4)$ .

g(x) = x + 1 for x ≠ 1, and g(1) = 4.Open in grapher →

Trace the graph toward $x = 1$ from either side: the heights approach $2$ . So $\displaystyle \lim_{x \to 1} g(x) = 2$ , even though $g(1) = 4$ . The solid dot is a distraction. The limit only cares about the path leading to $x = 1$ .

Common mistake

Don't read the limit off the solid dot. $\displaystyle \lim_{x \to a} f(x)$ and $f(a)$ are different questions. They can be equal, different, or one can exist without the other.

One-sided limits

Sometimes a function approaches one height from the left and a different height from the right. To describe each side separately, use one-sided limits:

$\displaystyle \lim_{x \to a^-} f(x)$ : the limit as $x$ approaches $a$ from the left (through values less than $a$ ).
$\displaystyle \lim_{x \to a^+} f(x)$ : the limit as $x$ approaches $a$ from the right (through values greater than $a$ ).

Here is a piecewise function:

h(x) = \begin{cases} x + 1, & x < 2 \\ x^2 - 3, & x \ge 2 \end{cases}

Left of 2 the graph heads to height 3; right of 2 it starts at height 1.Open in grapher →

From the left, $h(x) = x + 1$ approaches $2 + 1 = 3$ . From the right, $h(x) = x^2 - 3$ approaches $4 - 3 = 1$ . So

\lim_{x \to 2^-} h(x) = 3 \quad \text{and} \quad \lim_{x \to 2^+} h(x) = 1.

The two sides disagree, so there is no single number that $h(x)$ approaches. The two-sided limit does not exist.

When a two-sided limit exists

$\displaystyle \lim_{x \to a} f(x) = L$ exactly when both one-sided limits exist and equal $L$ :

\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = L.

Three ways a limit can fail to exist

Jump. The left and right limits are different numbers, as with $h$ above.
Unbounded behavior. The values grow without bound. Near $x = 0$ , $\dfrac{1}{x^2}$ takes values like $100$ , $10{,}000$ and $1{,}000{,}000$ . Since it approaches no real number, the limit does not exist. (You may see this written $\displaystyle \lim_{x \to 0} \frac{1}{x^2} = \infty$ , which describes how it fails: the values rise without bound.)
Oscillation. The values keep bouncing between different heights no matter how close you get. The classic example is $\sin\left(\dfrac{1}{x}\right)$ near $0$ : it swings between $-1$ and $1$ infinitely often.

Worked example: Estimating a limit with a table

Estimate $\displaystyle \lim_{x \to 0} \frac{\sin x}{x}$ , with $x$ in radians.

Solution. You can't plug in $x = 0$ , so build a table from both sides (values rounded).

$x$	$-0.1$	$-0.01$	$-0.001$	$0.001$	$0.01$	$0.1$
$\dfrac{\sin x}{x}$	$0.99833$	$0.99998$	$0.9999998$	$0.9999998$	$0.99998$	$0.99833$

The outputs creep toward $1$ from both sides, so $\displaystyle \lim_{x \to 0} \frac{\sin x}{x} = 1$ . This limit turns out to be one of the most important in calculus.

Worked example: Limits and values from a graph

Use the graph of $g$ in the section "Limits from graphs" to find $\displaystyle \lim_{x \to 3} g(x)$ , $g(3)$ , $\displaystyle \lim_{x \to 1} g(x)$ and $g(1)$ .

Solution. At $x = 3$ nothing unusual happens: the line reaches height $4$ and the point is filled in, so $\displaystyle \lim_{x \to 3} g(x) = 4$ and $g(3) = 4$ . At $x = 1$ the line heads to height $2$ , so the limit is $2$ , while the solid dot says $g(1) = 4$ .

Worked example: One-sided limits of a piecewise function

Let $p(x) = \begin{cases} 3 - x, & x < 1 \\ 2x, & x \ge 1. \end{cases}$ Find $\displaystyle \lim_{x \to 1^-} p(x)$ , $\displaystyle \lim_{x \to 1^+} p(x)$ and $\displaystyle \lim_{x \to 1} p(x)$ .

Solution. From the left, use $3 - x$ : it approaches $3 - 1 = 2$ . From the right, use $2x$ : it approaches $2(1) = 2$ . Both sides agree, so $\displaystyle \lim_{x \to 1} p(x) = 2$ . The formula changes at $x = 1$ , but the two pieces meet, so the limit exists.

Tip

When you build a table, use inputs from both sides and let them get closer to $a$ by factors of $10$ . If the two rows settle on the same number, that number is a strong estimate of the limit.

Practice

Practice 1

The table shows values of a function $f$ near $x = 3$ .

$x$	$2.9$	$2.99$	$2.999$	$3.001$	$3.01$	$3.1$
$f(x)$	$6.8$	$6.98$	$6.998$	$7.002$	$7.02$	$7.2$

Estimate $\displaystyle \lim_{x \to 3} f(x)$ .

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 2

Make a table of values to estimate $\displaystyle \lim_{x \to -1} \frac{x^2 - 1}{x + 1}$ .

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 3

A function $f$ is graphed below. Find $\displaystyle \lim_{x \to 2} f(x)$ .

The graph of f.Open in grapher →

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 4

For the same function $f$ in the previous problem, what is $f(2)$ ?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 5

Let $f(x) = \begin{cases} 2x - 1, & x < 3 \\ x + 2, & x \ge 3. \end{cases}$ Find $\displaystyle \lim_{x \to 3} f(x)$ .

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 6

What is $\displaystyle \lim_{x \to 0} \frac{|x|}{x}$ ?

Practice 7

Let $q(x) = \begin{cases} x^2, & x < 1 \\ 4 - x, & x \ge 1. \end{cases}$ Which statement is true?

Practice 8

Use a calculator and a table of values to estimate $\displaystyle \lim_{x \to 0} \frac{2^x - 1}{x}$ . Round to the nearest hundredth.

Enter a number. Fractions like 3/4 and sqrt(2) are OK.