L’Hospital’s Rule Questions and solutions

The L’Hospital’s Rule is mainly used in limit’s questions when the limits of some rational functions are indeterminate. Here is the worksheet on L’Hôpital’s rule with list of example limits problems for your practice and also solutions with understandable steps to learn how to use the L’Hospital’s rule to evaluate the limits of rational functions in calculus.

Evaluate $\displaystyle \large \lim_{x\,\to\,0}{\normalsize \dfrac{\sin{5x}-\sin{3x}}{x}}$

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

Evaluate $\displaystyle \large \lim_{x\,\to\,2}{\normalsize \dfrac{x^3-8}{x^2-4}}$

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

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

Evaluate $\displaystyle \large \lim_{x\,\to\,0}{\normalsize \dfrac{1-\cos{mx}}{1-\cos{nx}}}$

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

Evaluate $\displaystyle \large \lim_{x \,\to\, 3}{\normalsize \dfrac{\sqrt{3x}-3}{\sqrt{2x-4}-\sqrt{2}}}$

Evaluate $\displaystyle \large \lim_{x\,\to\,0}{\normalsize \dfrac{\log_{\displaystyle e}{\big(\cos{(\sin{x})}\big)}}{x^2}}$