# Limits Questions with Solutions

Once you are confident about the limit rules, you are ready to use them in the limits problems. The list of questions on limits with answers is given here for your practice. A worksheet with limits examples and solutions for you to learn how to evaluate the limits of the functions by the limits formulas in calculus.

## Limits methods

#### Direct substitution

#### Factorization

$(1).\,\,$ Evaluate $\displaystyle \large \lim_{x\,\to\,5}{\normalsize \dfrac{x^2-25}{x-5}}$

$(2).\,\,$ Evaluate $\displaystyle \large \lim_{x\,\to\,2}{\normalsize \dfrac{x^3+7x^2-36}{x^2+2x-8}}$

List of limits questions on factorisation to learn how to find the limit of a function by factoring.

#### Rationalization

$(1).\,\,$ Evaluate $\displaystyle \large \lim_{x\,\to\,0}{\normalsize \dfrac{\sqrt{1+x}-1}{x}}$

$(2).\,\,$ Evaluate $\displaystyle \large \lim_{x\,\to\,0}{\normalsize \dfrac{x}{\sqrt{a+x}-\sqrt{a-x}}}$

List of limit questions on rationalisation with solutions to learn how to find the limits by rationalization.

#### L’Hôpital’s Rule

d

$(1).\,\,$ Evaluate $\displaystyle \large \lim_{x\,\to\,-3}{\normalsize \dfrac{6+2x}{x^2+3x}}$

$(2).\,\,$ Evaluate $\displaystyle \large \lim_{x\,\to\,0}{\normalsize \dfrac{e^x-1-x}{x^2}}$

### Limit of functions

#### Algebraic functions

List of limit problems with solutions for the algebraic functions to find the limits of functions in which algebraic functions are involved.

#### Trigonometric functions

Evaluate $\displaystyle \large \lim_{x \,\to\, 0}{\normalsize \dfrac{x-\sin{x}}{x^3}}$

List of limit problems with solutions for the trigonometric functions to find the limits of functions in which trigonometric functions are involved.

#### Logarithmic functions

List of limit problems with solutions for the logarithmic functions to find the limits of functions in which logarithmic functions are involved.

#### Exponential functions

List of limit problems with solutions for the exponential functions to evaluate the limits of functions in which exponential functions are involved.

### Difficult Problems

List of hard/tough problems in limits with solutions.