How to Transform Limit Expressions That Preclude Using L’Hôpital’s Rule
Although L’Hôpital’s rule is a great shortcut for doing limit problems, you may sometimes be unable to proceed with a limit problem when substitution produces certain unacceptable forms.
For example, when substitution of the arrow-number into the limit expression produces one of the forms
you use the following logarithm trick to turn it into an acceptable form. Here’s how it works. Say you want to find
Substitution gives you
which equals 00, so you do the following:
Set the limit equal to y.
Take the log of both sides.
This limit is a
case, so tweak it.
Now you’ve got a
case, so you can use L’Hôpital’s rule.
The derivative of
and the derivative of
This is a
case, so use L’Hôpital’s rule again.
Hold your horses! This is not the answer.
Solve for y.
Do you see that the answer of 0 in Step 5 is the answer to the equation from way back in Step 2:
So, the 0 in Step 5 tells you that ln(y) = 0. Now solve for y:
Because you set your limit equal to y in Step 1, this, finally, is your answer:
Keep in mind that ordinary math doesn‘t work with infinity (or zero to the zero power). Don’t make the mistake of thinking that you can use ordinary arithmetic or the laws of exponents when dealing with any of the acceptable or unacceptable indeterminate forms. It might look like
should equal zero, for example, but it doesn’t. By the same token,