How to Transform Limit Problems in Order to Use 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 unacceptable forms,

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In these instances, you first have to tweak the problem to get an acceptable form before using L’Hôpital’s rule.

For instance, find

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Substituting infinity into x gives you

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so you’ve got to tweak it:

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Now you’ve got the

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case, so you’re all set to use L’Hôpital’s rule. The derivative of

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and the derivative of ex is ex, so

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Here’s another problem: What’s

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(Recall that

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means that x approaches 0 from the right only; this is a one-sided limit.) First, substitute zero into x (actually, since x is approaching zero from the right, you must imagine plugging a tiny positive number into x, or you can sort of think of it as plugging a “positive” zero into x). Substitution gives you

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which results in

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one of the unacceptable forms. So tweak the limit expression with some algebra:

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Now substitution gives you

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so you can finish with L’Hôpital’s rule:

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That’s it.