Loading [MathJax]/jax/element/mml/optable/BasicLatin.js

What is L'Hôpital's Rule?:
Practice Problems

Download this web page as a pdf with answer key
Problem 1

Use L'Hôpital's Rule to evaluate lim.

Problem 2

Use L'Hôpital's Rule to evaluate \displaystyle \lim_{x\to 0} \frac{\tan 2x}{3x}.

Problem 3

Use L'Hôpital's Rule to evaluate \displaystyle \lim_{x\to9}\frac{\sqrt x - 3}{x-9}

Problem 4

Use L'Hôpital's Rule to evaluate \displaystyle \lim_{x\to 8}\frac{2x^2-31x+120}{\sqrt{x+1}-3}

Problem 5

Use L'Hôpital's Rule to evaluate \displaystyle \lim_{x\to 5} \frac{3x^2 - 75}{1/5-1/x}

Problem 6

Use L'Hôpital's Rule to evaluate \displaystyle \lim_{x\to0} \frac{3x^2-2x}{3^x-5^x}

Problem 7

Evaluate \displaystyle \lim_{x\to0} \frac{\arctan(8x)}{\arcsin(2x)}

Problem 8

Evaluate \displaystyle \lim_{x\to0} \frac{x^2e^{0.5x}}{\tan^2(0.75x)}

Problem 9

Use L'Hôpital's Rule to evaluate \displaystyle \lim_{x\to\infty} \frac{4x^3+6x^2-9x+1}{2x^3-5x^2+8}

Problem 10

Use L'Hôpital's Rule to evaluate \displaystyle \lim\limits_{x\to\infty} \frac{e^{x^2}}{x^2+1}

Problem 11

Use L'Hôpital's Rule to evaluate \displaystyle \lim_{x\to\infty} \frac{e^{2x}-5x}{3x^2 + 8x}

Problem 12

Suppose \displaystyle f(x) = \frac{3\pi - 6 \arctan(9x)}{2\arctan(3x)-\pi}. Evaluate both of the following limits.

  1. \displaystyle \lim_{x\to\infty} f(x)
  2. \displaystyle \lim_{x\to-\infty} f(x)
Problem 13

Use L'Hôpital's Rule to evaluate \displaystyle \lim_{x\to 1/2^-} \frac{\tan \pi x}{\ln(1 - 2x)}.

Problem 14
  1. For any p>0 and any b>1, evaluate \displaystyle \lim_{x\to\infty} \frac{x^p}{b^x}.
  2. Explain the importance of your answer to part (a).
Problem 15
  1. For any p>0 and b>1, evaluate \displaystyle \lim_{x\to \infty} \frac{\log_b x}{x^p}.
  2. Explain the importance of your answer to part (a).
Return to lesson
Download this web page as a pdf with answer key

back to What is Cauchy's Extension of the Mean Value Theorem? next to When Does L'Hôpital's Rule Fail?

Ultimate Math Solver (Free)

Free Algebra Solver ... type anything in there!