L'Hôpital's Rule

You will probably have already considered limits in work on the sum of a geometric series, rational functions and asymptotes and differentiation from first principles. In this page, we will take the idea of limits further. L'Hôpital's rule gives us a technique to evaluate limits of indeterminate forms \(\large \frac{0}{0}\)and \(\large \frac{∞}{∞}\). The method for doing this is actually fairly straight forward, it requires you to be confident with differentiation techniques. In the examination, you might also be required to evaluate limits using Maclaurin Series. Check out that method on the page for Maclaurin Series.


Key Concepts

On this page, you should learn about

  • The indeterminate forms \(\large \frac{0}{0}\)and \(\large \frac{∞}{∞}\).
  • Evaluation of limits in the form \(\lim\limits_{x\rightarrow a} \frac{f(x)}{g(x)}\)

Summary

Test Yourself

Here is a quiz about the skills learned on this page



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Exam-style Questions

Question 1

Use l’Hôpital’s rule to determine the value of \(\large \lim\limits_{x\rightarrow 0}( x \ln x)\)

Hint

Full Solution

Question 2

Use l’Hôpital’s rule to determine the value of \(\large \lim\limits_{x\rightarrow 0}\frac{e^{x^2}-1}{\sin x^2}\)

Hint

Full Solution

Question 3

Use l’Hôpital’s rule to determine the value of \(\large \lim\limits_{x\rightarrow 0}\frac{1-\cos x^2}{x^4}\)

Hint

Full Solution

Question 4

Use l’Hôpital’s rule to determine the value of \(\large \lim\limits_{x\rightarrow 0}\frac{6\tan x-6x}{x^3}\)

Hint

Full Solution

MY PROGRESS

How much of L'Hôpital's Rule have you understood?