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Math 180

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Math 180

4.5 – Indeterminate Forms and L’Hôpital’s Rule

Let’s revisit limits for a moment. Consider the following limit:

As , the top and the bottom.

Limits of the type are said to be _______________.

Let’s revisit limits for a moment. Consider the following limit:

As , the top and the bottom.

Limits of the type are said to be _______________.

indeterminate

Previously, we would try to cancel a factor and then take the limit again. Now let’s look at another powerful tool for evaluating such indeterminate limits.

L’Hôpital’s Rule

Suppose that , that and are differentiable on an open interval containing , and that on if . Then,

Ex 1.

Use L’Hôpital’s Rule to find the following limits.

Note:L’Hôpital’s Rule also applies to one-sided limits as well (like ).

Type

L’Hôpital’s Rule also works for indeterminate forms of type .

Ex 2.

Find the following limits.

Type and

Sometimes we can use algebra to convert limits of type or into type or (then use L’Hôpital’s Rule).

Ex 3.

Find the following limits.

Type , , and

For these, try taking the logarithm of the function first, then take the limit, then exponentiate.

Ex 4.

Find the following limits.