L'Hopital's Rule

L'Hopital's Rule is a method for computing a limit of the form

c can be a number, , or . The conditions for applying it are:

1. The functions f and g are differentiable in an open interval containing c. (c may also be an endpoint of the open interval, if the limit is one-sided.)
1. g and are nonzero in the open interval, except possibly at c.
1. is defined, or is , or is .
1. As ,

If these conditions hold, then

In other words, f and g may be replaced by their derivatives.

Note that you're not applying the Quotient Rule to .

Example. Compute

Plugging into gives , so I can apply L'Hopital's Rule:

Example. Compute

As , , so I can apply L'Hopital's Rule:

Example. Compute

As , , so I {\it can't} apply L'Hopital's Rule. In fact, since the top and bottom are both positive,

Example. Compute

As , (which is not 0!). I convert the expression into a fraction by rationalizing:

As , , so I could apply L'Hopital's Rule. Instead, I'll divide the top and bottom by x:

Example. If you apply L'Hopital's Rule, and the limit you obtain is undefined, you may not conclude that the original limit is undefined. For example, consider

As , , so I can apply L'Hopital's Rule:

The last limit is undefined, because has no limit as . This implies that the 's in the reasoning above aren't valid. When you do a L'Hopital computation, the equalities are actually provisional, pending the existence of a limit in the chain.

In fact, the original limit exists:

Example. You can handle the indeterminate form by using algebra to convert the expression to a fraction, and then applying L'Hopital's Rule. Consider

As , . So

As , , so I can apply L'Hopital's Rule:

Example. The indeterminate form can be handled by taking logs, computing the limit using the techniques above, and finally exponentiating to undo the log. Consider

As , .

Let . Then

So

As , . So convert the expression to a fraction:

As , , so I can apply L'Hopital's Rule:

That is, . Therefore,

Example. Compute .

As , . Set . Take logs and simplify:

Take the limit as , applying L'Hopital's rule to the fraction:

Hence, .

Example. Compute .

This is an indeterminate form . Combine the fractions over a common denominator:

This is an form, so I can apply L'Hopital's Rule:

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Copyright 2005 by Bruce Ikenaga