So far, we have used l'Hopital's rule to find the limits of two indeterminate forms involving quotients.

There are several other indeterminate forms with which we can use this rule, such as those involving products, differences, and powers.

Here we have a limit of indeterminate form.

This is an indeterminate form involving a product.

In order to determine this limit using l'Hopital's rule, we first must convert it to an indeterminate form involving a quotient.

After taking the derivatives of the numerator and denominator, we then can evaluate the new form of the limit.

We can verify our answer by looking at a graph of the function. We see that the graph of the function approaches zero as x approaches zero, although the function is undefined at that point,

This is an example of a limit of indeterminate form involving a difference.

To find this limit, we first must convert it to an indeterminate quotient form.

Next, we apply l'Hopital's rule.

The new limit is still an indeterminate form involving a quotient.

Therefore, we must apply l'Hopital's rule a second time. Again, we take the derivatives of the numerator and denominator.

The limit is no longer in an indeterminate form. We now can evaluate it.

Now, let's look at a limit of indeterminate form involving a power. We cannot convert directly to a quotient and use l'Hopital's rule as in the previous examples. We must use a different approach.

First, let's just look at the function involved in the limit.

We can eliminate the exponential term by taking the logarithm of both sides of the expression.

Now, let's look at the limit of this new expression.

Using a trigonometric relationship, we can convert the expression to a quotient. This new limit is of an indeterminate form involving a quotient, so we can apply l'Hopital's rule.

Now, we have a limit that we can evaluate.

The value of this limit is not the answer to our original problem. This is the value of the limit of the logarithm of our original expression.

Using our knowledge of the relationship between natural logarithms and exponentials, we can write the expression in a different form.

Finally, substituting in our original expression, we arrive at the solution to the original limit.

What indeterminate form is this limit? Click the "Submit" button after selecting your answer.

What is the form of this new limit? Click the "Submit" button after selecting your answer.

What should we do next? Click the "Submit" button after selecting your answer.

What is the value of this limit? Click the "Submit" button after entering your answer.

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