Further Limits (inc l'Hôpital's Rule) (DP IB Maths: AA HL)

For each of the following limits, 

Consider the following limit:

Explain why it is appropriate to use l’Hôpital’s rule to attempt to evaluate this limit.

Use l’Hôpital’s rule to evaluate the limit.

Consider the function  defined by

Use l’Hôpital’s rule to evaluate .

Write down the value of

Hence write down the equation(s) of any horizontal asymptotes on the graph of giving a reason for your answer.

Show that  may also be evaluated without the use of l’Hôpital’s rule, and confirm that the limit found matches the answer from part (a)(i).

Determine the Maclaurin series for  in ascending powers of  up to the term in .

Consider the limit

Use Maclaurin series to evaluate the limit.

Use Maclaurin series to evaluate the limit

where is a non-zero real constant. Give your answer in terms of .

Use an appropriate method to evaluate the limit

Consider the function  defined by

where  is a constant. 

Given that  converges to a finite value, determine the value of .

Using the value of  found in part (a), use l’Hôpital’s rule to evaluate .

The first two terms, in ascending powers of , of the Maclaurin series for  are , where  are constants.

Write down the values of and , being sure to justify your answers.

For each of the following limits, 

if l’Hôpital’s rule may be used, then use the rule to evaluate the limit.

Consider the following limit:

Explain why it is appropriate to use l’Hôpital’s rule to attempt to evaluate this limit.

Use l’Hôpital’s rule to evaluate the limit.

Consider the function  defined by

Use l’Hôpital’s rule to evaluate .

Find .

Hence write down the equation(s) of any horizontal asymptotes on the graph of , giving a reason for your answer.

Show that  may also be evaluated without the use of l’Hôpital’s rule, and confirm that the limit found matches the answer from part (a)(i).

Find the Maclaurin series for , giving the terms and coefficients explicitly in ascending powers of up through the term in .

Consider the limit

Use Maclaurin series to evaluate the limit

where  are non-zero constants.  Give your answer in terms of  and  as appropriate.

Use an appropriate method to show that

for any .

Use the fact that    to evaluate the following limit:

Be sure to justify the validity of any method you use to determine the limit.