AHL 5.13—Limits and L’Hopitals Questionbank

Practice AHL 5.13—Limits and L’Hopitals with authentic IB Mathematics Analysis and Approaches (AA) exam questions for both SL and HL students. This question bank mirrors Paper 1, 2, 3 structure, covering key topics like functions and equations, calculus, complex numbers, sequences and series, and probability and statistics. Get instant solutions, detailed explanations, and build exam confidence with questions in the style of IB examiners.

Paper

Level

Question 1

HLPaper 1

Evaluate the limit of the function $g(x) = \frac{\ln(x)}{x - 1}$ as $x$ approaches 1.

Use L'Hôpital's Rule to find the limit of $g(x)$ as $x$ approaches 1.

[3]

Question 2

HLPaper 1

Consider the function $h(x) = \frac{e^{2x} - 1 - 2x}{x^2}$ .

Find the limit of $h(x)$ as $x$ approaches 0 using L'Hôpital's Rule.

[4]

Question 3

HLPaper 1

\lim _{x \rightarrow 0} \frac{\sin (x)-x}{x^3}

Using L'Hôpital's rule, show algebraically that the value of the limit is $-\frac{1}{6}$ .

[3]

Question 4

HLPaper 1

Use L'Hôpital's rule to find $\lim_{y \to 0} \frac{\arctan 2y}{\tan 3y}$ .

[5]

Question 5

HLPaper 1

Let $h(x) = \frac{1}{x^2}$ and $k(x) = \ln(x)$ .

Find the composite function $(h \circ k)(x)$ .

[1]

Determine the domain of the composite function $(h \circ k)(x)$ , given that $(h \circ k)(x) \neq \frac{1}{4}$ .

[5]

Evaluate the limit $\lim_{x \to 1} (h \circ k)(x)$ .

[2]

Now consider the function $f(x)=k(x)\sqrt{h(x)}$ . Hence, find $\int f(x) \, \mathrm{d}x$ .

[5]

Question 6

HLPaper 1

In a biological model for population growth under environmental stress, the growth rate is approximated by $\frac{\arctan (2 x)-2 x}{x^{3} \ln \left(1+x^{2}\right)}$ . Evaluate $\lim _{x \rightarrow 0} \frac{\arctan (2 x)-2 x}{x^{3} \ln \left(1+x^{2}\right)}$ .

Confirm the indeterminate form and apply l'Hôpital's rule three times.

[7]

Evaluate the limit using l'Hôpital's rule a fourth time.

[3]

Use Maclaurin series for $\arctan (2 x)$ and $\ln \left(1+x^{2}\right)$ to confirm the limit.

[3]

Question 7

HLPaper 1

Consider the limit $\lim _{x \rightarrow 0} \frac{e^{x}-1-x}{\sin ^{2} x}$ .

Verify that the limit is of indeterminate form and apply l'Hôpital's rule once.

[4]

Apply l'Hôpital's rule twice more to evaluate the limit.

[4]

Use the Maclaurin series for $e^{x}$ and $\sin x$ to confirm the limit.

[3]

Question 8

HLPaper 2

Consider $\lim _{x \rightarrow 0} \frac{\tan x-x}{x^{3}}$ .

Show the limit is indeterminate and apply l'Hôpital's rule twice.

[5]

Evaluate the limit using l'Hôpital's rule a third time.

[3]

Derive the Maclaurin series for $\tan x$ up to $x^{3}$ and use it to confirm the limit.

[3]

Sketch the graphs of $y=\tan x-x$ and $y=x^{3}$ for $x \in[-0.3,0.3]$ .

[2]

Question 9

HLPaper 2

Find the limit

\lim _{x \rightarrow 0} \frac{e^{2 x}-1-2 x}{x^{2} \tan x}

Show that the limit is indeterminate and apply l'Hôpital's rule once.

[3]

Apply l'Hôpital's rule again and evaluate the limit.

[4]

Plot the graphs of $y=e^{2 x}-1-2 x$ and $y=x^{2} \tan x$ for $x \in[-0.3,0.3]$ to confirm the limit behavior.

[2]

Question 10

HLPaper 2

Consider the limit

\lim _{x \rightarrow 0} \frac{x-\sin 3 x}{x^{3} \cos x} .

Show that the limit is of indeterminate form and apply l'Hôpital's rule once to simplify it.

[4]

Apply l'Hôpital's rule again to the result from part (a) and evaluate the limit.

[6]

Sketch the graphs of $y=x-\sin 3 x$ and $y=x^{3} \cos x$ for $x \in[-0.5,0.5]$ to illustrate the behavior of the function near $x=0$ .

[2]