AP Calculus BC : Euler's Method and L'Hopital's Rule

Study concepts, example questions & explanations for AP Calculus BC

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Example Questions

Example Question #3 : L'hospital's Rule

Calculate the following limit.

Possible Answers:

Correct answer:

Explanation:

If we plugged in  directly, we would get an indeterminate value of .

We can use L'Hopital's rule to fix this. We take the derivate of the top and bottom and reevaluate the same limit.

.

We still can't evaluate the limit of the new expression, so we do it one more time.

Example Question #4 : L'hospital's Rule

Find the 

.

Possible Answers:

Does Not Exist

Correct answer:

Explanation:

Subbing in zero into  will give you , so we can try to use L'hopital's Rule to solve.

First, let's find the derivative of the numerator. 

 is in the form , which has the derivative , so its derivative is 

 is in the form , which has the derivative , so its derivative is .

The derivative of  is  so the derivative of the numerator is .

In the denominator, the derivative of  is , and the derivative of  is . Thus, the entire denominator's derivative is .

Now we take the 

, which gives us 

Example Question #11 : Euler's Method And L'hopital's Rule

Evaluate the following limit:

Possible Answers:

Correct answer:

Explanation:

When you try to solve the limit using normal methods, you find that the limit approaches zero in the numerator and denominator, resulting in an indeterminate form "0/0". 

In order to evaluate the limit, we must use L'Hopital's Rule, which states that:

when an indeterminate form occurs when evaluting the limit.

Next, simply find f'(x) and g'(x) for this limit:

The derivatives were found using the following rules:

Next, using L'Hopital's Rule, evaluate the limit using f'(x) and g'(x):

Example Question #19 : L'hospital's Rule

Find the limit if it exists.

Hint: Apply L'Hospital's Rule.

Possible Answers:

Correct answer:

Explanation:

Through direct substitution, we see that the limit becomes

which is in indeterminate form.

As such we can use l'Hospital's Rule, which states that if the limit 

is in indeterminate form, then the limit is equivalent to

Taking the derivatives we use the power rule which states 

Using the power rule the limit becomes 

As such the limit exists and is

Example Question #20 : L'hospital's Rule

Find the limit if it exists.

Hint: Apply L'Hospital's Rule.

Possible Answers:

Correct answer:

Explanation:

Through direct substitution, we see that the limit becomes

which is in indeterminate form.

As such we can use l'Hospital's Rule, which states that if the limit 

is in indeterminate form, then the limit is equivalent to

Taking the derivatives we use the trigonometric rule which states 

 

where  is a constant.

Using l'Hospital's Rule we obtain

And through direct substitution we find

As such the limit exists and is

Example Question #23 : L'hospital's Rule

Find the limit:  

Possible Answers:

Correct answer:

Explanation:

By substituting the value of , we will find that this will give us the indeterminate form .  This means that we can use L'Hopital's rule to solve this problem.

L'Hopital states that we can take the limit of the fraction of the derivative of the numerator over the derivative of the denominator.  L'Hopital's rule can be repeated as long as we have an indeterminate form after every substitution.

Take the derivative of the numerator.

Take the derivative of the numerator.

Rewrite the limit and use substitution.

The limit is .

Example Question #481 : Ap Calculus Bc

Find the limit if it exists

Hint: Use L'Hospital's rule

 

 
Possible Answers:

Correct answer:

Explanation:

Directly evaluating for  yields the indeterminate form

we are able to apply L'Hospital's rule which states that if the limit is in indeterminate form when evaluated, then

As such the limit in the problem becomes

Evaluating for  yields

As such

 

and thus

Example Question #482 : Ap Calculus Bc

Evaluate  using L'hopital's rule.

Possible Answers:

Correct answer:

Explanation:

This important limit from elementary limit theory is usually proven using trigonometric arguments, but it can be shown using L'hopital's rule too.

 

Example Question #483 : Ap Calculus Bc

Evaluate the limit:

Possible Answers:

The limit does not exist

Correct answer:

Explanation:

When evaluating the limit using normal methods, we find that the indeterminate form  results. When this occurs, we must use L'Hopital's Rule, which states that for .

Taking the derivative of the top and bottom functions and evaluating the limit, we get

The derivatives were found using the following rules:

Example Question #484 : Ap Calculus Bc

Evaluate the limit:

Possible Answers:

The limit does not exist

Correct answer:

Explanation:

When evaluating the limit using normal methods, we find that we get the indeterminate form . When this occurs, we must use L'Hopital's Rule to evaluate the limit. The rule states that for the limits for which the indeterminate forms result,

Taking the derivatives for our limit, we get

The derivatives were found using the following rules:

 

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