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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:

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Math 180

Math 180

4.5 – Indeterminate Forms and L’Hôpital’s Rule


Math 180

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 _______________.


Math 180

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


Math 180

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.


Math 180

L’Hôpital’s the limit again. Now let’s look at another powerful tool for evaluating such indeterminate limits. Rule

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


Math 180

Ex 1. the limit again. Now let’s look at another powerful tool for evaluating such indeterminate limits.

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


Math 180

Note: the limit again. Now let’s look at another powerful tool for evaluating such indeterminate limits.L’Hôpital’s Rule also applies to one-sided limits as well (like ).


Math 180

Type the limit again. Now let’s look at another powerful tool for evaluating such indeterminate limits.

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

Ex 2.

Find the following limits.


Math 180

Type the limit again. Now let’s look at another powerful tool for evaluating such indeterminate limits. 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.


Math 180

Type the limit again. Now let’s look at another powerful tool for evaluating such indeterminate limits., , and

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

Ex 4.

Find the following limits.