1.1 Tables and graphs

Achievable AP Calculus AB

1. Limits

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Tables and graphs

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What you’ll learn

How to estimate limits with tables and graphs
One-sided limits
When a limit does or doesn’t exist

Limits are the foundational core of calculus. Calculus is the study of change, and limits are what allow us to analyze how a function behaves near a specific point, even if the function is undefined at the point.

Mathematically, a limit describes the value that a function’s output approaches as its input gets closer and closer to a target number.

Understanding limit notation

The limit of $f (x)$ as $x$ approaches $a$ is written as:

$x \to a lim f (x) = L$

This means that as $x$ gets closer to $a$ , the values of $f (x)$ get closer to $L$ .

Note that a limit is about what $f (x)$ is approaching, not necessarily what $f (x)$ equals at $x = a$ . The function value at $a$ might be different from $L$ , or it might not exist at all.

In fact, $f (a)$ does not have to equal $L$ or even be defined.

Estimating limits

There are three main ways to evaluate a limit:

Numerically (from a table)
Graphically (from a graph)
Analytically (using algebra or calculus)

This page focuses on the first two methods: estimating limits from tables and graphs.

Sidenote

On using Desmos

The AP exam now includes the built-in Desmos Graphing Calculator on calculator-active sections. Throughout this textbook, Desmos will be used to explore graphs and solve equations. Tips and shortcuts for using Desmos efficiently will also be included along the way to help you become more familiar with its features.

Limits from a table

One way to estimate a limit is by using a table of values. Choose $x$ -values that get closer and closer to $a$ from both sides, compute $f (x)$ , and look for a value that the outputs seem to settle toward.

Example 1. Estimate $x \to 1 lim f (x)$ based on the table below.

$x$ $f (x)$

$0.9$ $4.8$

$0.95$ $4.9$

$0.99$ $4.99$

$1.01$ $5.01$

$1.05$ $5.04$

$1.1$ $5.2$

$x$	$f (x)$
$0.9$	$4.8$
$0.95$	$4.9$
$0.99$	$4.99$
$1.01$	$5.01$
$1.05$	$5.04$
$1.1$	$5.2$

(spoiler)

As $x$ gets closer to $1$ , from either side, the values of $f (x)$ appear to get closer to $5$ , which suggests that

$x \to 1 lim f (x) = 5$

Example 2. Estimate the following limit by creating your own table of values.

$x \to 2 lim \frac{x ^{2} + x - 6}{x - 2} =$

(spoiler)

Even though $f (2)$ is undefined because the denominator is $0$ , a limit depends on the values near $x = 2$ . So we use values of $x$ close to $2$ :

$x$	$\frac{x ^{2} + x - 6}{x - 2}$
$1.9$	$4.9$
$1.99$	$4.99$
$1.999$	$4.999$
$2.001$	$5.001$
$2.01$	$5.01$
$2.1$	$5.1$

From the table, as $x$ approaches $2$ , the function values approach $5$ . So

$x \to 2 lim \frac{x ^{2} + x - 6}{x - 2} = 5$

This matches its graph - the curve approaches a $y$ -value of $5$ as $x$ approaches $2$ . Even though $f (2)$ is undefined, the limit exists.

Limits that do not exist

There are three common behaviors that lead to a limit that fails to exist. If $f (x)$ does any of the following as $x$ approaches $a$ , then the limit does not exist:

Oscillates or jumps between values
Approaches different numbers from the left and right sides (one-sided limits don’t match)
Increases or decreases without bound (to $\pm \infty$ )

One example of oscillating behavior is

$x \to 0 lim sin (\frac{1}{x})$

The table of values shows values of $x$ approaching $0$ :

$x$	$sin (\frac{1}{x})$
$- 0.1$	$0.544$
$- 0.01$	$0.506$
$- 0.001$	$- 0.827$
$- 0.0001$	$0.306$
$0.0001$	$- 0.306$
$0.001$	$0.827$
$0.01$	$- 0.506$
$0.1$	$- 0.544$

Despite the symmetry, the output values continue oscillating between positive and negative numbers instead of approaching a single number. Therefore, the limit does not exist.

One-sided limits

Sometimes a function behaves differently depending on whether $x$ approaches $a$ from the left or from the right. One-sided limits are used to describe these situations, with the following notation:

Left-hand limit:

$x \to a^{-} lim f (x)$

Notice the small ( $-$ ) above $a$ . This means $x$ approaches $a$ from the left (values smaller than $a$ ).

Right-hand limit:

$x \to a^{+} lim f (x)$

The small ( $+$ ) above $a$ means $x$ approaches $a$ from the right (values larger than $a$ ).

For a two-sided limit $x \to a lim f (x)$ to exist, both one-sided limits must exist and be equal.

If the one-sided limits are different, then the limit does not exist (DNE).

Based on the table, estimate

$x \to 4 lim f (x)$

$x$ $f (x)$

$3.9$ $- 2.19$

$3.99$ $- 2.199$

$3.999$ $- 2.1999$

$4.001$ $5.001$

$4.01$ $5.01$

$4.1$ $5.1$

$x$	$f (x)$
$3.9$	$- 2.19$
$3.99$	$- 2.199$
$3.999$	$- 2.1999$
$4.001$	$5.001$
$4.01$	$5.01$
$4.1$	$5.1$

(spoiler)

As $x \to 4$ from the left (values smaller than $4$ ), $f (x)$ appears to approach $- 2.2$ , suggesting

$x \to 4^{-} lim f (x) = - 2.2$

As $x \to 4$ from the right (values greater than $4$ ), $f (x)$ appears to approach $5$ , suggesting

$x \to 4^{+} lim f (x) = 5$

Because the one-sided limits don’t match, there is no single value that $f (x)$ approaches as $x \to 4$ . Therefore, $x \to 4 lim f (x)$ does not exist.

Limits from a graph

A graph gives a visual way to track what $f (x)$ does as $x$ approaches $a$ . You can see whether the function approaches a single height, shoots upward or downward without bound, jumps to a different value, or oscillates.

To determine a limit from a graph, follow these steps:

Follow the graph toward $x = a$ from both the left and the right.
If $f (x)$ approaches the same height from both sides, that height is the limit - even if there is a hole at that point or the function’s actual value is somewhere else.
If $f (x)$ approaches different heights from either side, the one-sided limits are different, so the limit does not exist (DNE).
- If $f (x)$ increases or decreases without bound, the limit fails to exist in the usual sense but the behavior can still be described using $\infty$ or $- \infty$ .

The graph of $f$ is shown below with a vertical asymptote at $x = - 4$ . Based on the graph, find the following limits:

Graph of f

a) $x \to - 4 lim f (x)$

b) $x \to 0 lim f (x)$

c) $f (2)$

d) $x \to 2 lim f (x)$

Answers

(spoiler)

a) Does not exist
b) $- 1$
c) $3$
d) Does not exist

Solutions

a) $x \to - 4 lim f (x)$

(spoiler)

From the left:

As $x$ gets closer to $- 4$ from the left, $f (x)$ increases without bound toward $\infty$ . So

$x \to - 4^{-} lim f (x) = \infty$

From the right:

As $x$ gets closer to $- 4$ from the right, $f (x)$ decreases without bound toward $- \infty$ . So

$x \to - 4^{+} lim f (x) = - \infty$

Since the one-sided limits don’t match, the overall limit $x \to - 4 lim f (x)$ does not exist.

b) $x \to 0 lim f (x)$

(spoiler)

Although $f (0)$ is undefined (a hole indicated by the open circle), the limit depends on what the graph approaches as $x$ gets close to $0$ . From both sides, the graph approaches $y = - 1$ . So

$x \to 0 lim f (x) = - 1$

c) $f (2)$

(spoiler)

At $x = 2$ , the filled-in dot corresponds to $y = 3$ , so the point $(2, 3)$ is included in the function. Therefore,

$f (2) = 3$

d) $x \to 2 lim f (x)$

(spoiler)

To find the limit, observe how the graph approaches $x = 2$ .

From the left:

As $x \to 2^{-}$ , the graph approaches $y = 1$ , so

$x \to 2^{-} lim f (x) = 1$

From the right:

As $x \to 2^{+}$ , the graph approaches $y = 3$ , so

$x \to 2^{+} lim f (x) = 3$

Because these one-sided limits are different, $x \to 2 lim f (x)$ does not exist.

Tables and graphs

What you’ll learn

How to estimate limits with tables and graphs
One-sided limits
When a limit does or doesn’t exist

Mathematically, a limit describes the value that a function’s output approaches as its input gets closer and closer to a target number.

Understanding limit notation

The limit of $f (x)$ as $x$ approaches $a$ is written as:

$x \to a lim f (x) = L$

This means that as $x$ gets closer to $a$ , the values of $f (x)$ get closer to $L$ .

Note that a limit is about what $f (x)$ is approaching, not necessarily what $f (x)$ equals at $x = a$ . The function value at $a$ might be different from $L$ , or it might not exist at all.

In fact, $f (a)$ does not have to equal $L$ or even be defined.

Estimating limits

There are three main ways to evaluate a limit:

Numerically (from a table)
Graphically (from a graph)
Analytically (using algebra or calculus)

This page focuses on the first two methods: estimating limits from tables and graphs.

Sidenote

On using Desmos

Limits from a table

Example 1. Estimate $x \to 1 lim f (x)$ based on the table below.

$x$ $f (x)$

$0.9$ $4.8$

$0.95$ $4.9$

$0.99$ $4.99$

$1.01$ $5.01$

$1.05$ $5.04$

$1.1$ $5.2$

$x$	$f (x)$
$0.9$	$4.8$
$0.95$	$4.9$
$0.99$	$4.99$
$1.01$	$5.01$
$1.05$	$5.04$
$1.1$	$5.2$

(spoiler)

As $x$ gets closer to $1$ , from either side, the values of $f (x)$ appear to get closer to $5$ , which suggests that

$x \to 1 lim f (x) = 5$

Example 2. Estimate the following limit by creating your own table of values.

$x \to 2 lim \frac{x ^{2} + x - 6}{x - 2} =$

(spoiler)

Even though $f (2)$ is undefined because the denominator is $0$ , a limit depends on the values near $x = 2$ . So we use values of $x$ close to $2$ :

$x$	$\frac{x ^{2} + x - 6}{x - 2}$
$1.9$	$4.9$
$1.99$	$4.99$
$1.999$	$4.999$
$2.001$	$5.001$
$2.01$	$5.01$
$2.1$	$5.1$

From the table, as $x$ approaches $2$ , the function values approach $5$ . So

$x \to 2 lim \frac{x ^{2} + x - 6}{x - 2} = 5$

This matches its graph - the curve approaches a $y$ -value of $5$ as $x$ approaches $2$ . Even though $f (2)$ is undefined, the limit exists.

Limits that do not exist

There are three common behaviors that lead to a limit that fails to exist. If $f (x)$ does any of the following as $x$ approaches $a$ , then the limit does not exist:

Oscillates or jumps between values
Approaches different numbers from the left and right sides (one-sided limits don’t match)
Increases or decreases without bound (to $\pm \infty$ )

One example of oscillating behavior is

$x \to 0 lim sin (\frac{1}{x})$

The table of values shows values of $x$ approaching $0$ :

$x$	$sin (\frac{1}{x})$
$- 0.1$	$0.544$
$- 0.01$	$0.506$
$- 0.001$	$- 0.827$
$- 0.0001$	$0.306$
$0.0001$	$- 0.306$
$0.001$	$0.827$
$0.01$	$- 0.506$
$0.1$	$- 0.544$

Despite the symmetry, the output values continue oscillating between positive and negative numbers instead of approaching a single number. Therefore, the limit does not exist.

One-sided limits

Left-hand limit:

$x \to a^{-} lim f (x)$

Notice the small ( $-$ ) above $a$ . This means $x$ approaches $a$ from the left (values smaller than $a$ ).

Right-hand limit:

$x \to a^{+} lim f (x)$

The small ( $+$ ) above $a$ means $x$ approaches $a$ from the right (values larger than $a$ ).

For a two-sided limit $x \to a lim f (x)$ to exist, both one-sided limits must exist and be equal.

If the one-sided limits are different, then the limit does not exist (DNE).

Based on the table, estimate

$x \to 4 lim f (x)$

$x$ $f (x)$

$3.9$ $- 2.19$

$3.99$ $- 2.199$

$3.999$ $- 2.1999$

$4.001$ $5.001$

$4.01$ $5.01$

$4.1$ $5.1$

$x$	$f (x)$
$3.9$	$- 2.19$
$3.99$	$- 2.199$
$3.999$	$- 2.1999$
$4.001$	$5.001$
$4.01$	$5.01$
$4.1$	$5.1$

(spoiler)

As $x \to 4$ from the left (values smaller than $4$ ), $f (x)$ appears to approach $- 2.2$ , suggesting

$x \to 4^{-} lim f (x) = - 2.2$

As $x \to 4$ from the right (values greater than $4$ ), $f (x)$ appears to approach $5$ , suggesting

$x \to 4^{+} lim f (x) = 5$

Because the one-sided limits don’t match, there is no single value that $f (x)$ approaches as $x \to 4$ . Therefore, $x \to 4 lim f (x)$ does not exist.

Limits from a graph

To determine a limit from a graph, follow these steps:

Follow the graph toward $x = a$ from both the left and the right.
If $f (x)$ approaches the same height from both sides, that height is the limit - even if there is a hole at that point or the function’s actual value is somewhere else.
If $f (x)$ approaches different heights from either side, the one-sided limits are different, so the limit does not exist (DNE).
- If $f (x)$ increases or decreases without bound, the limit fails to exist in the usual sense but the behavior can still be described using $\infty$ or $- \infty$ .

The graph of $f$ is shown below with a vertical asymptote at $x = - 4$ . Based on the graph, find the following limits:

a) $x \to - 4 lim f (x)$

b) $x \to 0 lim f (x)$

c) $f (2)$

d) $x \to 2 lim f (x)$

Answers

(spoiler)

a) Does not exist
b) $- 1$
c) $3$
d) Does not exist

Solutions

a) $x \to - 4 lim f (x)$

(spoiler)

From the left:

As $x$ gets closer to $- 4$ from the left, $f (x)$ increases without bound toward $\infty$ . So

$x \to - 4^{-} lim f (x) = \infty$

From the right:

As $x$ gets closer to $- 4$ from the right, $f (x)$ decreases without bound toward $- \infty$ . So

$x \to - 4^{+} lim f (x) = - \infty$

Since the one-sided limits don’t match, the overall limit $x \to - 4 lim f (x)$ does not exist.

b) $x \to 0 lim f (x)$

(spoiler)

Although $f (0)$ is undefined (a hole indicated by the open circle), the limit depends on what the graph approaches as $x$ gets close to $0$ . From both sides, the graph approaches $y = - 1$ . So

$x \to 0 lim f (x) = - 1$

c) $f (2)$

(spoiler)

At $x = 2$ , the filled-in dot corresponds to $y = 3$ , so the point $(2, 3)$ is included in the function. Therefore,

$f (2) = 3$

d) $x \to 2 lim f (x)$

(spoiler)

To find the limit, observe how the graph approaches $x = 2$ .

From the left:

As $x \to 2^{-}$ , the graph approaches $y = 1$ , so

$x \to 2^{-} lim f (x) = 1$

From the right:

As $x \to 2^{+}$ , the graph approaches $y = 3$ , so

$x \to 2^{+} lim f (x) = 3$

Because these one-sided limits are different, $x \to 2 lim f (x)$ does not exist.

Achievable AP Calculus AB

1. Limits

Our AP Calculus AB course is now in "early access" - get 50% off for a limited time.