What is L'Hopital's rule?

L'Hopital's rule is a mathematical tool that helps us find the value of certain limits that might otherwise be difficult to determine. It's especially useful when we have fractions where the numerator and denominator both seem to approach zero or infinity. We can often get an answer by taking the derivatives (which show how things change) of the numerator and denominator of the fraction and then evaluating the limit again. This rule is a valuable tool for solving tricky math problems involving limits.

Formula

L'Hopital's rule enables us to evaluate certain indeterminate forms by taking the derivatives of the numerator and denominator and then finding the limit of their ratio.

Suppose we have a limit of the form $\lim_{{x \to c}} \frac{f(x)}{g(x)}$where both $f(x)$ and $g(x)$ approach zero or infinity as $x$ approaches $c$ (i.e., $lim_{{x \to c}} f(x) = \lim_{{x \to c}} g(x) = 0$ or $\lim_{{x \to c}} f(x) = \lim_{{x \to c}} g(x) = \pm \infty$.

If the limit of the ratio of their derivatives exists as $x$ approaches $c$, i.e,

In other words, to find the value of the original limit, we can take the derivatives of both the numerator and denominator and then evaluate the new expression's limit.

Forms

L'Hopital's rule is capable of dealing with different limit forms that would otherwise be challenging to solve using other methods. Here are a few examples of such limit forms:

Examples

We'll look at a few examples and understand how L'Hopital's rule helps us find limits in tricky situations where other methods might not work.

Example 1

Question: Evaluate the limit $\lim_{{x \to 3}} \frac{x^2 + x - 12}{x^2 - 9}$

You can see graphically that there exists a value, but mathematically if we put $x=3$ we'll face $\frac{0}{0}$problem.

Answer: If we try to solve it normally by putting $x=3$into the above expression, we will end up with $\lim_{{x \to 3}} \frac{3^2 + 3 - 12}{3^2 - 9} = \lim_{{x \to 3}} \frac{0}{0}$ which means we cannot further evaluate this expression. Next, we will try to use L'hopital's rule to see how it helps us to solve the limit.

Now try to put $x=3$ into the new expression and see if it is solvable.

In the example above, we have seen the case $\lim_{{x \to c}} \frac{0}{0}$ where L'hopital's rule helps us to calculate the limit by derivating the numerator and denominator and evaluating the limit on the new expression.

Example 2

Question: Evaluate the limit $\lim_{{x \to \infty}} \frac{3x^2 + 2x}{4x^2 - 5x}$

You can see graphically that the graph approaches to a value as $x$ approaches positive infinity, but mathematically if we put $x=\infin$ we'll face $\frac{\infin}{\infin}$problem.

Answer: At first glance, it seems like an indeterminate form $\frac{\infty}{\infty}$, where both the numerator and denominator grow without bounds. However, by applying L'hopital’s rule, we can find the limit.

Applying L'Hopital’s rule:

Now, as $x$ approaches infinity, the fraction becomes:

We have another indeterminate form. We can apply L'hopital’s rule again:

Applying L'Hopital’s rule for the second time:

Quiz

Now that you know the L'Hopital's rule, you can challenge yourself with a quiz.

Conclusion

L'Hopital's rule is a valuable mathematical tool that aids in evaluating limits involving indeterminate forms​. By taking derivatives of the numerator and denominator and re-evaluating the limit, we can often obtain results.