Limit Examples: 44 Problems with Answers

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Limit.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

The value a function gets closer and closer to as the input approaches a specific target value, without necessarily reaching it.

What output do you get closer and closer to as you get closer and closer to some input?

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: A limit names the single output a function approaches as the input closes in on a target, whether or not the function is defined there.

Common stuck point: The procedure for limit is the easy part; the trap is just substituting $a$ and stopping when you get $\frac{0}{0}$ . Asking "Am I asked what the output heads toward as the input closes in, rather than the output exactly at that input?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I asked what the output heads toward as the input closes in, rather than the output exactly at that input?

Worked Examples

Example 1

easy

Find

\lim_{x \to 3} (2 x + 1)

Answer

\lim_{x \to 3} (2 x + 1) = 7

First step

Since

2x + 1

is a polynomial, it is continuous everywhere.

Full solution

2
For continuous functions, we can evaluate the limit by direct substitution.
3
Substitute $x = 3$ : $2(3) + 1 = 6 + 1 = 7$ .

When a function is continuous at a point, the limit equals the function value at that point. Polynomials are continuous everywhere, so direct substitution always works for polynomial limits.

Example 2

medium

Find

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

Example 3

hard

Find

\lim_{x \to 0} \frac{\sin x}{x}

Example 4

medium

Evaluate

\lim_{x \to 4} \frac{\sqrt{x} - 2}{x - 4}

Example 5

medium

Evaluate

\lim_{x \to 0} \frac{\tan x}{x}

Example 6

medium

Evaluate

\lim_{x \to 1} \frac{x^3 - 1}{x - 1}

Example 7

hard

Evaluate

\lim_{x \to \infty} \left(\sqrt{x^2 + x} - x\right)

Example 8

hard

Evaluate

\lim_{x \to 0} \frac{\ln(1 + x)}{x}

Example 9

hard

Use the squeeze theorem to evaluate

\lim_{x \to 0} x^2 \sin(1/x)

Example 10

challenge

Evaluate

\lim_{x \to 0} \frac{\tan x - \sin x}{x^3}

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

Find

\lim_{x \to 5} (x^{2} - 3 x + 2)

Example 2

medium

Find

\lim_{x \to - 1} \frac{x^{2} + 3 x + 2}{x + 1}

Example 3

easy

Evaluate

\lim_{x \to 3} (2x + 1)

Example 4

easy

Evaluate

\lim_{x \to 2} x^2

Example 5

easy

Evaluate

\lim_{x \to 0} (5 - x)

Example 6

easy

Evaluate

\lim_{x \to 4} \sqrt{x}

Example 7

easy

Evaluate

\lim_{x \to 1} \frac{x+3}{x+1}

Example 8

easy

Find

\lim_{x \to 2^+} (x - 2)

and state whether it matches the left-hand limit.

Example 9

easy

Evaluate

\lim_{x \to 0} \cos x

Example 10

easy

Evaluate

\lim_{x \to 5} 7

Example 11

medium

Evaluate

\lim_{x \to 3} \frac{x^2 - 9}{x - 3}

Example 12

medium

Evaluate

\lim_{x \to 0} \frac{x^2 + 2x}{x}

Example 13

medium

Evaluate

\lim_{x \to 4} \frac{x - 4}{\sqrt{x} - 2}

Example 14

medium

Evaluate

\lim_{x \to 0} \frac{\sin x}{x}

Example 15

medium

Evaluate

\lim_{x \to \infty} \frac{3x^2 + 1}{x^2 + 5}

Example 16

medium

Evaluate

\lim_{x \to \infty} \frac{2x + 1}{x^2 + 3}

Example 17

medium

Evaluate

\lim_{x \to 2} \frac{x^2 - 5x + 6}{x - 2}

Example 18

medium

For

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

, find

\lim_{x \to 2} f(x)

Example 19

challenge

Evaluate

\lim_{x \to 0} \frac{\sqrt{1 + x} - 1}{x}

Example 20

challenge

Evaluate

\lim_{x \to \infty} \left( \sqrt{x^2 + x} - x \right)

Example 21

challenge

Evaluate

\lim_{x \to 0} \frac{1 - \cos x}{x^2}

Example 22

medium

Evaluate

\lim_{x \to 1} \frac{x^3 - 1}{x - 1}

Example 23

easy

Evaluate

\lim_{x \to 0} (x^2 + 3x + 4)

Example 24

easy

Evaluate

\lim_{x \to 3} \frac{x^2 - 9}{x - 3}

Example 25

easy

Evaluate

\lim_{x \to 0} \frac{x}{x + 1}

Example 26

easy

Evaluate

\lim_{x \to 0} e^x

Example 27

medium

Evaluate

\lim_{x \to \infty} \frac{3x^2 + 5}{2x^2 - x + 1}

Example 28

medium

Evaluate

\lim_{x \to 0} \frac{1 - \cos x}{x^2}

Example 29

medium

Evaluate

\lim_{x \to 0^+} \frac{1}{x}

Example 30

medium

Evaluate

\lim_{x \to \infty} \frac{5x + 3}{x^2 + 1}

Example 31

medium

Evaluate

\lim_{h \to 0} \frac{(2 + h)^2 - 4}{h}

Example 32

hard

Evaluate

\lim_{x \to 0} \frac{e^x - 1}{x}

Example 33

hard

Evaluate

\lim_{x \to 0} \left(1 + 2x\right)^{1/x}

Example 34

hard

Evaluate

\lim_{x \to \infty} \left(1 + \tfrac{3}{x}\right)^x

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Related Concepts

Function Derivative Types of Continuity and Discontinuity

Background Knowledge

These ideas may be useful before you work through the harder examples.

function definition