1.1 Tables and graphs

Achievable AP Calculus AB

1. Limits

Our AP Calculus AB course is currently in development and is a work-in-progress.

Tables and graphs

9 min read

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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 one of the core ideas in calculus. Calculus is all about describing change, and limits let us talk about what a function is doing near a point - even if the function isn’t defined exactly at that point.

A limit describes what a function’s output approaches as its input (usually $x$ ) gets closer to a particular value.

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 $L$ .

A key point: 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.

1. From a table

One way to estimate a limit is by using a table of values. You 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.

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

$x$ $f (x)$

$0.9$ $4.8$

$0.95$ $4.9$

$0.99$ $4.99$

$1.01$ $5.01$

$1.05$ $5.04$

$1.1$ $5.2$

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

Solution

(spoiler)

As $x$ gets closer to $1$ from both sides, the values of $f (x)$ get closer to $5$ . So we estimate

$x \to 1 lim f (x) \approx 5.$

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

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

Sidenote

On using Desmos

Since the AP exam is now administered on the Bluebook app with Desmos Graphing Calculator built into certain sections, it’s highly recommended you use it to see how different functions behave and confirm the solutions to many of these problems. Practicing with it will help you become more familiar with the features. Some tips with the notation and tricks will also be included throughout the textbook to help you use it efficiently.

Solution

(spoiler)

Because $(x - 2)$ is in the denominator, the function is not defined at $x = 2$ , so $f (2)$ is undefined. But a limit depends on the function’s behavior near $2$ , not the value at $2$ .

Create a table using $x$ -values that get closer and closer to $2$ from both sides.

To do this in Desmos, type in the function

$\frac{x ^{2} + x - 6}{x - 2}$

Click the gear icon (Edit List) that’s above and to the right of the input box, and select the “Create Table” icon.

Enter $x$ -values, and Desmos will display the corresponding $y$ -values. For example:

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

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 what you’ll see on the graph in Desmos: the curve approaches a $y$ -value of $5$ as $x \to 2$ . Even though $f (2)$ is undefined, the limit exists.

With trickier functions, a table may not show a clear pattern.

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

Solution

Make a table of values close to $x = 0$ , such as:

$x$	$f (x)$
$- 0.01$	$0.506$
$- 0.001$	$- 0.827$
$- 0.0001$	$0.306$
$0.0001$	$- 0.306$
$0.001$	$0.827$
$0.01$	$- 0.506$

These values don’t settle toward a single number. As $x$ gets closer to $0$ , the expression $\frac{1}{x}$ becomes very large in magnitude, and $sin (\frac{1}{x})$ keeps oscillating between $- 1$ and $1$ .

Because the function does not approach one specific value, the limit does not exist (DNE).

One-sided limits

Sometimes a function behaves differently depending on whether you approach $a$ from the left or from the right. In that case, it helps to use 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). One-sided limits come up often with piecewise functions, step functions, and jump discontinuities, and they also matter later when you study derivatives and integrals.

Example

Based on the table, estimate

$x \to 4 lim f (x)$

$x$ $f (x)$

$3.8$ $- 2.16$

$3.9$ $- 2.19$

$3.99$ $- 2.199$

$4.01$ $5.001$

$4.2$ $5.1$

$4.4$ $5.13$

$x$	$f (x)$
$3.8$	$- 2.16$
$3.9$	$- 2.19$
$3.99$	$- 2.199$
$4.01$	$5.001$
$4.2$	$5.1$
$4.4$	$5.13$

Solution

(spoiler)

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

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

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

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

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

2. 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 $f (x)$ shoots off to $\pm \infty$ , you can also write “DNE” next to $\infty$ or $- \infty$ to show the limit is not a finite value.

Find the following limits based on the graph of $f (x)$ shown below:

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 $\infty$ ). You may write

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

From the right:

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

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

The one-sided behaviors don’t match, and neither side approaches a finite number, so 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 (there’s an 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 is at $y = 3$ , 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, look at what the graph approaches near $x = 2$ .

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.

When limits don’t exist

There are three common behaviors that lead to a limit that doesn’t exist. If $f (x)$ does any of the following as $x$ approaches $a$ :

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 $\pm \infty$ )

then the limit does not exist, and you should write “DNE” as your answer.

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

A limit describes what a function’s output approaches as its input (usually $x$ ) gets closer to a particular value.

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 $L$ .

A key point: 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.

1. From a table

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

$x$ $f (x)$

$0.9$ $4.8$

$0.95$ $4.9$

$0.99$ $4.99$

$1.01$ $5.01$

$1.05$ $5.04$

$1.1$ $5.2$

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

Solution

(spoiler)

As $x$ gets closer to $1$ from both sides, the values of $f (x)$ get closer to $5$ . So we estimate

$x \to 1 lim f (x) \approx 5.$

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

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

Sidenote

On using Desmos

Solution

(spoiler)

Because $(x - 2)$ is in the denominator, the function is not defined at $x = 2$ , so $f (2)$ is undefined. But a limit depends on the function’s behavior near $2$ , not the value at $2$ .

Create a table using $x$ -values that get closer and closer to $2$ from both sides.

To do this in Desmos, type in the function

$\frac{x ^{2} + x - 6}{x - 2}$

Click the gear icon (Edit List) that’s above and to the right of the input box, and select the “Create Table” icon.

Enter $x$ -values, and Desmos will display the corresponding $y$ -values. For example:

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

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 what you’ll see on the graph in Desmos: the curve approaches a $y$ -value of $5$ as $x \to 2$ . Even though $f (2)$ is undefined, the limit exists.

With trickier functions, a table may not show a clear pattern.

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

Solution

Make a table of values close to $x = 0$ , such as:

$x$	$f (x)$
$- 0.01$	$0.506$
$- 0.001$	$- 0.827$
$- 0.0001$	$0.306$
$0.0001$	$- 0.306$
$0.001$	$0.827$
$0.01$	$- 0.506$

Because the function does not approach one specific value, the limit does not exist (DNE).

One-sided limits

Sometimes a function behaves differently depending on whether you approach $a$ from the left or from the right. In that case, it helps to use 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.

Example

Based on the table, estimate

$x \to 4 lim f (x)$

$x$ $f (x)$

$3.8$ $- 2.16$

$3.9$ $- 2.19$

$3.99$ $- 2.199$

$4.01$ $5.001$

$4.2$ $5.1$

$4.4$ $5.13$

$x$	$f (x)$
$3.8$	$- 2.16$
$3.9$	$- 2.19$
$3.99$	$- 2.199$
$4.01$	$5.001$
$4.2$	$5.1$
$4.4$	$5.13$

Solution

(spoiler)

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

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

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

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

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

2. 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 $f (x)$ shoots off to $\pm \infty$ , you can also write “DNE” next to $\infty$ or $- \infty$ to show the limit is not a finite value.

Find the following limits based on the graph of $f (x)$ shown below:

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 $\infty$ ). You may write

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

From the right:

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

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

The one-sided behaviors don’t match, and neither side approaches a finite number, so 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 (there’s an 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 is at $y = 3$ , 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, look at what the graph approaches near $x = 2$ .

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.

When limits don’t exist

There are three common behaviors that lead to a limit that doesn’t exist. If $f (x)$ does any of the following as $x$ approaches $a$ :

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 $\pm \infty$ )

then the limit does not exist, and you should write “DNE” as your answer.

Achievable AP Calculus AB

1. Limits

Our AP Calculus AB course is currently in development and is a work-in-progress.

Tables and graphs

9 min read

Font

Discuss

Feedback

What you’ll learn:

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

A limit describes what a function’s output approaches as its input (usually $x$ ) gets closer to a particular value.

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 $L$ .

A key point: 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.

1. From a table

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

$x$ $f (x)$

$0.9$ $4.8$

$0.95$ $4.9$

$0.99$ $4.99$

$1.01$ $5.01$

$1.05$ $5.04$

$1.1$ $5.2$

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

Solution

(spoiler)

As $x$ gets closer to $1$ from both sides, the values of $f (x)$ get closer to $5$ . So we estimate

$x \to 1 lim f (x) \approx 5.$

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

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

Sidenote

On using Desmos

Solution

(spoiler)

Because $(x - 2)$ is in the denominator, the function is not defined at $x = 2$ , so $f (2)$ is undefined. But a limit depends on the function’s behavior near $2$ , not the value at $2$ .

Create a table using $x$ -values that get closer and closer to $2$ from both sides.

To do this in Desmos, type in the function

$\frac{x ^{2} + x - 6}{x - 2}$

Click the gear icon (Edit List) that’s above and to the right of the input box, and select the “Create Table” icon.

Enter $x$ -values, and Desmos will display the corresponding $y$ -values. For example:

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

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 what you’ll see on the graph in Desmos: the curve approaches a $y$ -value of $5$ as $x \to 2$ . Even though $f (2)$ is undefined, the limit exists.

With trickier functions, a table may not show a clear pattern.

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

Solution

Make a table of values close to $x = 0$ , such as:

$x$	$f (x)$
$- 0.01$	$0.506$
$- 0.001$	$- 0.827$
$- 0.0001$	$0.306$
$0.0001$	$- 0.306$
$0.001$	$0.827$
$0.01$	$- 0.506$

Because the function does not approach one specific value, the limit does not exist (DNE).

One-sided limits

Sometimes a function behaves differently depending on whether you approach $a$ from the left or from the right. In that case, it helps to use 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.

Example

Based on the table, estimate

$x \to 4 lim f (x)$

$x$ $f (x)$

$3.8$ $- 2.16$

$3.9$ $- 2.19$

$3.99$ $- 2.199$

$4.01$ $5.001$

$4.2$ $5.1$

$4.4$ $5.13$

$x$	$f (x)$
$3.8$	$- 2.16$
$3.9$	$- 2.19$
$3.99$	$- 2.199$
$4.01$	$5.001$
$4.2$	$5.1$
$4.4$	$5.13$

Solution

(spoiler)

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

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

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

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

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

2. 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 $f (x)$ shoots off to $\pm \infty$ , you can also write “DNE” next to $\infty$ or $- \infty$ to show the limit is not a finite value.

Find the following limits based on the graph of $f (x)$ shown below:

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 $\infty$ ). You may write

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

From the right:

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

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

The one-sided behaviors don’t match, and neither side approaches a finite number, so 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 (there’s an 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 is at $y = 3$ , 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, look at what the graph approaches near $x = 2$ .

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.

When limits don’t exist

There are three common behaviors that lead to a limit that doesn’t exist. If $f (x)$ does any of the following as $x$ approaches $a$ :

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 $\pm \infty$ )

then the limit does not exist, and you should write “DNE” as your answer.

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

A limit describes what a function’s output approaches as its input (usually $x$ ) gets closer to a particular value.

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 $L$ .

A key point: 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.

1. From a table

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

$x$ $f (x)$

$0.9$ $4.8$

$0.95$ $4.9$

$0.99$ $4.99$

$1.01$ $5.01$

$1.05$ $5.04$

$1.1$ $5.2$

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

Solution

(spoiler)

As $x$ gets closer to $1$ from both sides, the values of $f (x)$ get closer to $5$ . So we estimate

$x \to 1 lim f (x) \approx 5.$

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

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

Sidenote

On using Desmos

Solution

(spoiler)

Because $(x - 2)$ is in the denominator, the function is not defined at $x = 2$ , so $f (2)$ is undefined. But a limit depends on the function’s behavior near $2$ , not the value at $2$ .

Create a table using $x$ -values that get closer and closer to $2$ from both sides.

To do this in Desmos, type in the function

$\frac{x ^{2} + x - 6}{x - 2}$

Click the gear icon (Edit List) that’s above and to the right of the input box, and select the “Create Table” icon.

Enter $x$ -values, and Desmos will display the corresponding $y$ -values. For example:

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

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 what you’ll see on the graph in Desmos: the curve approaches a $y$ -value of $5$ as $x \to 2$ . Even though $f (2)$ is undefined, the limit exists.

With trickier functions, a table may not show a clear pattern.

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

Solution

Make a table of values close to $x = 0$ , such as:

$x$	$f (x)$
$- 0.01$	$0.506$
$- 0.001$	$- 0.827$
$- 0.0001$	$0.306$
$0.0001$	$- 0.306$
$0.001$	$0.827$
$0.01$	$- 0.506$

Because the function does not approach one specific value, the limit does not exist (DNE).

One-sided limits

Sometimes a function behaves differently depending on whether you approach $a$ from the left or from the right. In that case, it helps to use 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.

Example

Based on the table, estimate

$x \to 4 lim f (x)$

$x$ $f (x)$

$3.8$ $- 2.16$

$3.9$ $- 2.19$

$3.99$ $- 2.199$

$4.01$ $5.001$

$4.2$ $5.1$

$4.4$ $5.13$

$x$	$f (x)$
$3.8$	$- 2.16$
$3.9$	$- 2.19$
$3.99$	$- 2.199$
$4.01$	$5.001$
$4.2$	$5.1$
$4.4$	$5.13$

Solution

(spoiler)

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

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

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

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

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

2. 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 $f (x)$ shoots off to $\pm \infty$ , you can also write “DNE” next to $\infty$ or $- \infty$ to show the limit is not a finite value.

Find the following limits based on the graph of $f (x)$ shown below:

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 $\infty$ ). You may write

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

From the right:

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

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

The one-sided behaviors don’t match, and neither side approaches a finite number, so 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 (there’s an 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 is at $y = 3$ , 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, look at what the graph approaches near $x = 2$ .

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.

When limits don’t exist

There are three common behaviors that lead to a limit that doesn’t exist. If $f (x)$ does any of the following as $x$ approaches $a$ :

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 $\pm \infty$ )

then the limit does not exist, and you should write “DNE” as your answer.