Section 3.6

Although we study 7 different indeterminate forms in this section, L’Hopital’s Rule can only be used on limits of the form

.

On the next test I will put one limit that is not a L’Hopital’s Rule problem to make sure you are actually checking to see if L’Hopital’s Rule applies before using it. See the warning to the left of the solution to example 2b.

Remember that although we use L’Hopital’s Rule on quotients, we do not use the quotient rule when finding derivatives. The numerator and denominator are treated as separate entities.

If the new limit you get after using L’Hopital’s Rule is still zero over zero or infinity over infinity, you can apply L’Hopital’s rule again. However, if you use L’Hopital’s Rule several times and see no end in sight, try using some Algebra or Trigonometry to rewrite your limit in a different form.

Here is the method we use for limits of the type

1. Let y equal the expression you are trying to take the limit of.

2. Take the natural logarithm of both sides of the equation.

3. Use the property of logarithms that allows you to pull the exponent down in front of the logarithm.

4. If possible, simplify. Usually there is no simplification to be done.

5. Take the limit of both sides of the equation.

6. Find the limit on the right side of the equation using the methods of this section.

7. Perform the e^x function on both sides of the equation.

8. Simplify the left side to get the limit of y.

9. Replace y with what it was originally equal to.