How to Transform Limit Expressions That Preclude Using L'Hôpital's Rule - dummies

How to Transform Limit Expressions That Preclude Using L’Hôpital’s Rule

By Mark Ryan

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

image0.png

you use the following logarithm trick to turn it into an acceptable form. Here’s how it works. Say you want to find

image1.png

Substitution gives you

image2.png

which equals 00, so you do the following:

  1. Set the limit equal to y.

    image3.png

  2. Take the log of both sides.

    image4.png

  3. This limit is a

    image5.png

    case, so tweak it.

    image6.png

  4. Now you’ve got a

    image7.png

    case, so you can use L’Hôpital’s rule.

    The derivative of

    image8.png

    and the derivative of

    image9.png

  5. This is a

    image10.png

    case, so use L’Hôpital’s rule again.

    image11.png

    Hold your horses! This is not the answer.

  6. 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:

    image12.png

    So, the 0 in Step 5 tells you that ln(y) = 0. Now solve for y:

    image13.png

    Because you set your limit equal to y in Step 1, this, finally, is your answer:

    image14.png

Keep in mind that ordinary math doesnt 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

image15.png

should equal zero, for example, but it doesn’t. By the same token,

image16.png