Limits using l'Hôpital's Rule & Maclaurin Series (DP IB Analysis & Approaches (AA): HL): Exam Questions

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.

$\underset{x\to 0}{\lim }\frac{{\mathrm{e}}^x-1}{\sin x}$

$\underset{x\to 0}{\lim }\frac{\cos x}{1-{\mathrm{e}}^x}$

$\underset{x\to \frac{\mathrm{\pi }}{2}}{\lim }\frac{1-\cot 2x}{\mathrm{cosec}x\sec x}$

Consider the following limit:

$\underset{x\to 0}{\lim }\frac{\cos x+2\sin x-\sin 2x-1}{1-\cos x}$

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

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

Consider the function $f$ defined by

$f\left(x\right)=\frac{2+5{\mathrm{e}}^{x^2}}{9{\mathrm{e}}^{x^2}-7}$

(i) Use l’Hôpital’s rule to find $\underset{x\to +\infty }{\lim }f\left(x\right)$.

(ii) Write down the value of $\underset{x\to -\infty }{\lim }f\left(x\right)$.

Hence write down the equation(s) of any horizontal asymptotes on the graph of $y=f\left(x\right),$ giving a reason for your answer.

Show that $\underset{x\to +\infty }{\lim }f\left(x\right)$ 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 $\ln \left(1+x^2\right)$, up to and including the term in $x^6$.

Consider the limit

$\underset{x\to 0}{\lim }\frac{\ln \left(1+x^2\right)+\cos x-1}{x^2}$

Use Maclaurin series to find the limit.

(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).

Consider the function $f$ defined by

 $f\left(x\right)=\frac{p-\arctan \left(\cos x\right)}{x^2}$ 

where $p\in \mathrm{ℝ}$ is a constant. 

Given that $\underset{x\to 0}{\lim }f\left(x\right)$ converges to a finite value, determine the value of $p$.

Using the value of $p$ found in part (a), use l’Hôpital’s rule to find $\underset{x\to 0}{\lim }f\left(x\right)$.

The first two terms, in ascending powers of $x$, of the Maclaurin series for $\arctan \left(\cos x\right)$ are $q+r{x}^2$, where $q,r\in \mathrm{ℝ}$ are constants.

Hence, or otherwise, find the values of $q$ and $r$.

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.

$\underset{x\to 0}{\lim }\frac{x{\left(x+1\right)}^{17}}{{\left(x+1\right)}^{17}-1}$

$\underset{x\to 0}{\lim }\frac{\cos x}{\arctan x}$

$\underset{x\to +\infty }{\lim }\frac{\ln x}{\sqrt{x}}$

Consider the following limit:

$\underset{x\to 0}{\lim }\frac{{\mathrm{e}}^{x^2}-1}{1-\cos 3x}$ 

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

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

Consider the function $f$ defined by

$f\left(x\right)=\frac{9-4{\mathrm{e}}^x}{3{\mathrm{e}}^x+7}$

(i) Use l’Hôpital’s rule to find $\underset{x\to +\infty }{\lim }f\left(x\right)$.

(ii) Find $\underset{x\to -\infty }{\lim }f\left(x\right)$.

Hence write down the equation(s) of any horizontal asymptotes on the graph of $y=f\left(x\right)$, giving a reason for your answer.

Show that $\underset{x\to +\infty }{\lim }f\left(x\right)$ 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 $\arctan \left(\sin x\right)$, up to and including the term in $x^5$.

Consider the limit

$\underset{x\to 0}{\lim }\frac{\arctan \left(\sin x\right)}{x}$

Use Maclaurin series to find the limit.

(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).

Use an appropriate method to show that

$\underset{x\to +\infty }{\lim }\frac{x^n}{{\mathrm{e}}^x}=0$ 

for any $n\in \mathrm{\mathbb{N}}$.

Use the fact that  $x=\frac{1}{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$x$}\right.}$  to evaluate the following limit:

 $\underset{x\to 0^{+}}{\lim }x\ln x$ 

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