In this section, you will learn what L'Hopital's rule is, and how to use it. You will see that some problems require using L'Hopital's rule more than once. Before attempting this section, you should be excellent at finding derivatives.

In this section, you will learn what L'Hopital's rule is, and how to use it. You will see that some problems require using L'Hopital's rule more than once. Before attempting this section, you should be excellent at finding derivatives.

Typeset step-by-step solutions.

\(\displaystyle \lim_{x \to 0}\left(\dfrac{e^x - 1}{x}\right)\)

\(1\)

\(\displaystyle \lim_{x \to 0}\left(\dfrac{e^x - 1}{x^3}\right)\)

\(\infty\)

\(\displaystyle \lim_{x \to 0}\left(\dfrac{2x^2}{\cos x - 1}\right)\)

\(-4\)

\(\displaystyle \lim_{x \to 1}\left(\dfrac{x^9 - 1}{x^5 - 1}\right)\)

\(\dfrac{9}{5}\)

\(\displaystyle\lim_{x \to \infty}\dfrac{x^2}{e^x}\)

\(0\)

\(\displaystyle\lim_{x \to \infty}\dfrac{\ln x}{x}\)

\(0\)

\(\displaystyle\lim_{x \to 0}\dfrac{\sin(7x)}{\sin(4x)}\)

\(\dfrac{7}{4}\)

\(\displaystyle\lim_{x \to 0}\dfrac{\sin(8x)}{3x}\)

\(\dfrac{8}{3}\)

\(\displaystyle\lim_{x \to \infty}x\ln x\)

\(\infty\)

\(\displaystyle\lim_{x \to \infty}x^{1 / x}\)

\(1\)

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

\(e^{-2}\)

\(\displaystyle\lim_{x \to 0} \dfrac{\sin x}{x}\)

\(1\)

Conclude by giving your students these challenges:

- Jugs of Wine by NRICH
- What Does Random Look Like? by NRICH
- Point of Contact Triangle

Cubist Cuts by NRICH: The first part of this challenge also appears in Mathematics, A Human Endeavor by Harold R. Jacobs, on page 31. One of their diagrams is helpful, and doesn't appear on the NRICH page.

A farmer has 36 pieces of fence. He wants to arrange them into a rectangle or a regular polygon. Which shapes can he make? This problem is based on Jo Boaler's 36 Fences problem, which can be found on page 51 of Jo Boaler's *Studies in Mathematical Thinking.*