Practice Limit in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

Worksheet PDF Free Answer-key PDF Family

Quick 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?

Showing a random 20 of 50 problems.

Example 1

medium

Evaluate

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

Example 2

easy

Evaluate

\lim_{x \to 5} 7

Example 3

medium

Evaluate

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

Example 4

medium

Evaluate

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

Example 5

easy

Find

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

and state whether it matches the left-hand limit.

Example 6

medium

Evaluate

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

Example 7

medium

Evaluate

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

Example 8

easy

Evaluate

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

Example 9

medium

Find

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

Example 10

easy

Evaluate

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

Example 11

challenge

Evaluate

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

Example 12

medium

Evaluate

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

Example 13

medium

Evaluate

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

Example 14

hard

Evaluate

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

Example 15

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 16

hard

Evaluate

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

Example 17

medium

Evaluate

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

Example 18

medium

Find

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

Example 19

medium

Evaluate

\lim_{x \to 0} \frac{\sin 5x}{x}

Example 20

challenge

Evaluate

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

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Related Concepts

Function Derivative Types of Continuity and Discontinuity