AP®MathsCollege BoardCalculus BCStudy GuidesUnit 1: Limits & ContinuityLimitsEvaluating Limits Numerically & Graphically

Evaluating Limits Numerically & Graphically (College Board AP® Calculus BC): Study Guide

Written by: Roger B

Reviewed by: Dan Finlay

Updated on 19 May 2026

Limits from tables

How can I estimate a limit using values in a table?

Values of a function in a table can show the behavior of a function near a point
- This can allow you to estimate the limit at that point
For example, let $f$ be the function defined by $f (x) = \frac{1 - \cos x}{x^{2}}$
- The table below shows values of the function near $x = 0$
- Note that the function is not defined at $x = 0$ , because $f (0) = \frac{0}{0}$

x	f(x)
-0.1	0.49958347
-0.01	0.49999583
-0.001	0.49999996
0	not defined
0.001	0.49999996
0.01	0.49999583
0.1	0.49958347

From the table we can see that $f (x)$ gets nearer and nearer to 0.5 as $x$ gets nearer and nearer to 0
- Therefore we can estimate that $\lim_{x \to 0} f (x)$ is equal to 0.5
- Analytical methods would need to be used to confirm that this is indeed the limit

Examiner Tips and Tricks

In your exam, you might be asked to verify the value limit using a table. However, if you are asked to find the value of a limit, you should use an analytical method.

Limits from graphs

How can I find limits using a graph with given points?

To find $\lim_{x \to a^{-}} f (x)$
- Look at the $y$ -values of points on the graph just before $x = a$
To find $\lim_{x \to a^{+}} f (x)$
- Look at the $y$ -values of points on the graph just after $x = a$
Remember $\lim_{x \to a} f (x)$ does not necessarily equal $f (a)$
- Don't look at the point $x = a$
- Look at the points just before and after $x = a$

Graph of piecewise function with a jump at x = −2 and x = 2, showing f(−2) = 2, limit at −2 is −1, limit at 2 is 3, right-hand limit is 1. — **Example of limits from a graph**

How can I estimate a limit using a graph?

A graph can show the behavior of a function near a point
- This can allow you to estimate the limit at that point
For example, let $f$ be the function defined by $f (x) = \frac{1 - \cos x}{x^{2}}$
- The graph below shows the behavior of the function near $x = 0$
- Note that the function is not defined at $x = 0$ , because $f (0) = \frac{0}{0}$

A graph of the function y=(1-cosx)/x^2 between x=-2 and x=2, showing limiting behavior as x approaches zero — **Example of estimating a limit using a graph**

From the graph we can see that $f (x)$ gets nearer and nearer to 0.5 as $x$ gets nearer and nearer to 0
- Therefore we can estimate that $\lim_{x \to 0} f (x)$ is equal to 0.5
- Analytical methods would need to be used to confirm that this is indeed the limit

Examiner Tips and Tricks

You can graph functions on your graphing calculator to check your answers when determining limits analytically.

Worked Example

Piecewise graph of f: a curve from x = −4 to near 2 with open point at (-2,1) and closed point at (2, 6), then a line segment from (2,4) open to (4,2) closed. There is also an isolated point at (-2, 3).

The figure above shows the graph of the function $f$ .

Use the graph to find the following limits or state they do not exist:

(a) $\lim_{x \to - 2} f (x)$

(b) $\lim_{x \to 2^{+}} f (x)$

Answer:

(a)

Ignore the point when $x = - 2$ and look at the graph just before and just after this point

As $x \to - 2$ from both sides, $f (x) \to 1$

Both parts of the graph are heading towards the point (-2, 1)

Therefore, this is the limit even though the graph does not go through that point

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

(b)

Look at the graph just after the point $x = 2$

As $x \to 2$ from the right, $f (x) \to 4$

The graph is heading towards the point (2, 4)

Therefore, this is the right-hand limit even though the graph does not go through that point

$\lim_{x \to 2^{+}} f (x) = 4$

(c)

Ignore the point when $x = 2$ and look at the graph just before and just after this point

As $x \to 2$ from the left, $f (x) \to 6$

The graph is heading towards the point (2, 6)

$\lim_{x \to 2^{-}} f (x) = 6$

The right-hand side limit is equal to 4 using part (b)

Therefore, the left-hand limit and the right-hand limit are not equal

$\lim_{x \to 2} f (x)$ does not exist

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Evaluating Limits Numerically & Graphically (College Board AP® Calculus BC): Study Guide

Limits from tables

How can I estimate a limit using values in a table?

Examiner Tips and Tricks

Limits from graphs

How can I find limits using a graph with given points?

How can I estimate a limit using a graph?

Examiner Tips and Tricks

Worked Example

Unlock more, it's free!

Join the 100,000+ Students that ❤️ Save My Exams

Author: Roger B

Reviewer: Dan Finlay