L'Hospital's Rule (College Board AP® Calculus AB): Study Guide

Note that if substitution gives limits that look like or , these are not indeterminate forms, and L'Hospital's rule cannot be used

• In the first case, , the limit will just be equal to

• In the second case, , the limit will diverge to either or depending on the behavior near the limit point

  • See the 'Infinite Limits & Limits at Infinity' study guide

Other limit methods will also sometimes work when substitution gives an indeterminate form

• For example algebraic simplification, multiplying by conjugates or multiplying by reciprocals

  • See the 'Evaluating Limits Analytically' study guide

Before beginning to use L'Hospital's rule to evaluate a limit

• Be sure to confirm that using substitution gives an indeterminate form

• Otherwise L'Hospital's rule is not valid

Use L’Hospital’s rule to evaluate each of the following limits:

(a)

Answer:

This limit could also be found by 'multiplying by reciprocals', but the question says to use L'Hospital's rule

First check that substitution gives an indeterminate form, so L'Hospital's rule is valid

which is an indeterminate form

Find the derivatives of the numerator and denominator

Apply L'Hospital's Rule,

There's no x in that final expression, so x going to infinity doesn't matter!

(b)

Answer:

First check that substitution gives an indeterminate form, so L'Hospital's rule is valid

which is an indeterminate form

Find the derivatives of the numerator and denominator

Apply L'Hospital's Rule,

That is another indeterminate form, so we can repeat the process again

Apply L'Hospital's rule again

And that is also an indeterminate form, so we can repeat the process again

Apply L'Hospital's rule again

That is not an indeterminate form, so that's the answer we're looking for

Simplify the fraction