DP IB Maths: AA HL

Topic Questions

5.12 Further Limits (inc l'Hôpital's Rule)

1a
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4 marks

For each of the following limits, 

(i)
determine whether or not l’Hôpital’s rule may be used to evaluate the limit, giving a reason for your answer;  and

(ii)
if l’Hôpital’s rule may be used, then use the rule to evaluate the limit.
limit as x rightwards arrow 0 of fraction numerator e to the power of x minus 1 over denominator sin space x end fraction
1b
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2 marks
limit as x rightwards arrow 0 of fraction numerator cos space x over denominator 1 minus e to the power of x end fraction
1c
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5 marks
limit as x rightwards arrow straight pi over 2 of fraction numerator 1 minus cot space 2 x over denominator cosec space x space sec space x end fraction

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2a
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2 marks

Consider the following limit:

 limit as x rightwards arrow 0 of fraction numerator cos space x plus 2 space sin space x minus sin space 2 x minus 1 over denominator 1 minus cos space x end fraction 

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

2b
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4 marks

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

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3a
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4 marks

Consider the function f defined by

f open parentheses x close parentheses equals fraction numerator 2 plus 5 e to the power of x squared end exponent over denominator 9 e to the power of x squared end exponent minus 7 end fraction 

Use l’Hôpital’s rule to evaluate limit as x rightwards arrow plus infinity of f open parentheses x close parentheses .

Write down the value oflimit as x rightwards arrow negative infinity of f open parentheses x close parentheses

3b
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2 marks

Hence write down the equation(s) of any horizontal asymptotes on the graph of y equals f open parentheses x close parentheses comma giving a reason for your answer.

3c
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3 marks

Show that limit as x rightwards arrow plus infinity of f open parentheses x close parentheses 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).

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4a
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2 marks

Determine the Maclaurin series for ln open parentheses 1 plus x squared close parentheses in ascending powers of x up to the term in x to the power of 6.

 

4b
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5 marks

Consider the limit

limit as x rightwards arrow 0 of fraction numerator ln open parentheses 1 plus x squared close parentheses plus cos space x minus 1 over denominator x squared end fraction

Use Maclaurin series to evaluate the limit.

4c
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5 marks
(i)
Show that it would also be appropriate to use l’Hôpital’s rule to attempt to evaluate the limit.

(ii)
Evaluate the limit using l’Hôpital’s rule, and confirm that this matches your answer in part (b).

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5
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5 marks

Use Maclaurin series to evaluate the limit

 limit as x rightwards arrow 0 of fraction numerator sin open parentheses k x squared close parentheses over denominator x squared end fraction 

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

 

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6
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6 marks

Use an appropriate method to evaluate the limit

limit as x rightwards arrow 0 of fraction numerator x minus x space cos space 3 x over denominator x cubed end fraction

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7a
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2 marks

Consider the function f defined by

 f open parentheses x close parentheses equals fraction numerator p minus arctan open parentheses cos space x close parentheses over denominator x squared end fraction 

where p element of straight real numbers is a constant. 

Given that limit as x rightwards arrow 0 of f open parentheses x close parentheses converges to a finite value, determine the value of p.

7b
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6 marks

Using the value of p found in part (a), use l’Hôpital’s rule to evaluate limit as x rightwards arrow 0 of f open parentheses x close parentheses.

7c
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3 marks

The first two terms, in ascending powers of x, of the Maclaurin series for arctan open parentheses cos space x close parentheses are q plus r x squared, where q comma space r element of straight real numbers are constants.

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

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1a
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4 marks

For each of the following limits,

(i)
determine whether or not l’Hôpital’s rule may be used to evaluate the limit, giving a reason for your answer;  and

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

 limit as x rightwards arrow 0 of space fraction numerator sin space x over denominator x squared plus 2 x end fraction
1b
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2 marks
limit as x rightwards arrow 0 of space fraction numerator cos space x over denominator x squared plus 2 x end fraction
1c
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5 marks
limit as x rightwards arrow straight pi over 2 of space fraction numerator sec space x over denominator 1 minus tan space x end fraction

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2a
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2 marks

Consider the following limit: 

limit as x rightwards arrow 0 of space fraction numerator negative 1 plus cos space 2 x over denominator x squared end fraction 

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

2b
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2 marks

Show that employing l’Hôpital’s rule once leads to an indeterminate form when you attempt to evaluate the limit.

2c
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2 marks

By employing l’Hôpital’s rule a second time, show that the limit exists and find its value.

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3a
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3 marks

Consider the function f defined by

space f left parenthesis x right parenthesis space equals fraction numerator 7 minus 3 x over denominator 12 x plus 5 end fraction

Use l’Hôpital’s rule to evaluate limit as x rightwards arrow infinity of space f left parenthesis x right parenthesis.

3b
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2 marks

Hence write down the equation(s) of any horizontal asymptotes on the graph of y equals f left parenthesis x right parenthesis comma giving a reason for your answer.

3c
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4 marks

Show that  f left parenthesis x right parenthesis  may be rewritten in the form 

f left parenthesis x right parenthesis space fraction numerator 7 over x minus 3 over denominator 12 plus 5 over x end fraction

Hence show that  limit as x rightwards arrow infinity of space f left parenthesis x right parenthesis  may also be evaluated without the use of l’Hôpital’s rule. 

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4a
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2 marks

By substituting negative x into the Maclaurin series for e to the power of x, determine the Maclaurin series for e to the power of negative x end exponent.

4b
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5 marks

Consider the limit

limit as x rightwards arrow 0 of fraction numerator e to the power of x minus e to the power of negative x end exponent over denominator 2 x end fraction 

Use Maclaurin series to evaluate the limit.

4c
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4 marks
(i)
Show that it would also be appropriate to use l’Hôpital’s rule to attempt to evaluate the limit. 

(ii)
Evaluate the limit using l’Hôpital’s rule, and confirm that this matches your answer in part (b).

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5a
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3 marks

Find the Maclaurin series for cos space 2 x.

5b
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4 marks

Hence evaluate the limit

 

 limit as x rightwards arrow 0 of space fraction numerator 1 minus cos space 2 x over denominator x squared end fraction

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6
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5 marks

Use an appropriate method to evaluate the limit

 begin mathsize 16px style limit as x rightwards arrow 0 of fraction numerator sin space x minus x over denominator x cubed end fraction end style

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1a
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4 marks

For each of the following limits, 

(i)
determine whether or not l’Hôpital’s rule may be used to evaluate the limit, giving a reason for your answer; and 

(ii)
Indented content here...


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

limit as x rightwards arrow 0 of fraction numerator x open parentheses x plus 1 close parentheses to the power of 17 over denominator open parentheses x plus 1 close parentheses to the power of 17 minus 1 end fraction
1b
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2 marks
limit as x rightwards arrow 0 of fraction numerator cos space x over denominator arctan space x end fraction
1c
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4 marks
limit as x rightwards arrow plus infinity of fraction numerator ln space x over denominator square root of x end fraction

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2a
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2 marks

Consider the following limit:

limit as x rightwards arrow 0 of fraction numerator e to the power of x squared end exponent minus 1 over denominator 1 minus cos space 3 x end fraction 

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

2b
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5 marks

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

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3a
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5 marks

Consider the function f defined by

f open parentheses x close parentheses equals fraction numerator 9 minus 4 e to the power of x over denominator 3 e to the power of x plus 7 end fraction 

Use l’Hôpital’s rule to evaluate limit as x rightwards arrow plus infinity of f open parentheses x close parentheses.

Find limit as x rightwards arrow negative infinity of f open parentheses x close parentheses.

3b
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3 marks

Hence write down the equation(s) of any horizontal asymptotes on the graph of y equals f open parentheses x close parentheses, giving a reason for your answer.

3c
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3 marks

Show that limit as x rightwards arrow plus infinity of f open parentheses x close parentheses 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).

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4a
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5 marks

Find the Maclaurin series for arctan open parentheses sin space x close parentheses, giving the terms and coefficients explicitly in ascending powers of x up through the term in x to the power of 5.

4b
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3 marks

Consider the limit

limit as x rightwards arrow 0 of fraction numerator arctan open parentheses sin space x close parentheses over denominator x end fraction 

Use Maclaurin series to evaluate the limit.
4c
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4 marks
(i)
Show that it would also be appropriate to use l’Hôpital’s rule to attempt to evaluate the limit.

(ii)
Evaluate the limit using l’Hôpital’s rule, and confirm that this matches your answer in part (b).

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5
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6 marks

Use Maclaurin series to evaluate the limit

limit as x rightwards arrow 0 of fraction numerator sin space p x plus x space cos to the power of 2 space end exponent q x over denominator x end fraction 

where p comma space q space element of straight real numbers are non-zero constants.  Give your answer in terms of p and q as appropriate.

 

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6
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6 marks

Use an appropriate method to show that

limit as x rightwards arrow plus infinity of x to the power of n over e to the power of x equals 0 

for any n element of straight natural numbers.

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7
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5 marks

Use the fact that  x equals fraction numerator 1 over denominator bevelled 1 over x end fraction  to evaluate the following limit:

 limit as x rightwards arrow 0 to the power of plus of x space ln space x 

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

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