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 $h(x) = \frac{e^{2x} - 1 - 2x}{x^2}$ as $x$ approaches 0.

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 ${\lim_{{y\to 0}}} \frac{{\text{arctan}\ 2y}}{{\tan\ 3y}}$ .

[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 $(h \circ k)(x)$ with range $y\in\R,y\not=\frac{1}{4}$

[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 $\int\! f(x)dx$ .

[5]

Question 6

HLPaper 2

Consider a model of stock prices where the price $P(t)$ at time $t$ is given by the function $P(t) = e^{-2t} \cdot \ln(t^2 + 1)$ . Let $V(t)=\frac{d}{dt}P(t)$ represent the volatility of the price at a given moment in time.

Initially, there was no volatility. Then for a few moments in time volatility had increased before becoming 0 again at $t_1$ . Find the value of $t_1$ .

[2]

Find the point in time where the price is equal to volatility.

[1]

As $t\rightarrow \infty$ , what happens to the price?

[6]

Question 7

HLPaper 1

Consider the function $f(x) = \frac{e^x - 1}{x}$ for $x \neq 0$ and $f(0) = 1$ .

Show that $\lim_{x \to 0} f(x)$ exists and find its value.

[3]

Evaluate $\lim_{x \to 0} \frac{\ln(1+x)}{x}$ using L'Hôpital's Rule.

[3]

Find the limit $\lim_{x \to 0} \frac{e^{2x} - 1}{x}$ using L'Hôpital's Rule.

[3]

Consider the limit $\lim_{x \to \infty} \frac{x}{e^x}$ . Show that this limit is equal to 0 using L'Hôpital's Rule.

[3]

Evaluate $\lim_{x \to 0} \frac{\sin(x)}{x}$ without using L'Hôpital's Rule.

[3]

Question 8

HLPaper 1

Consider the function $f(x) = \frac{e^x - 1}{x}$ as $x$ approaches 0.

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

[2]

Question 9

HLPaper 1

Determine the limit of the function $h(x) = \frac{x^2 - 4}{x - 2}$ as $x$ approaches 2.

Find the limit of $h(x)$ as $x$ approaches 2 using L'Hôpital's Rule.

[3]

Question 10

HLPaper 1

The function $h$ is defined by $h(y) = e^y \sin y$ , where $y \in \mathbb{R}$ .

The function $k$ is defined by $k(y) = e^y \cos y$ , where $y \in \mathbb{R}$ .

Show that $k(y)$ satisfies the equation $k''(y) = 2(k'(y) - k(y))$ .

[4]

Hence, deduce that $k^{(4)}(y)=2\left(k^{\prime \prime \prime}(y)-k^{\prime \prime}(y)\right)$ .

[1]

Using the result from part 2, find the Maclaurin series for $k(y)$ up to and including the $y^4$ term.

[5]

Find the Maclaurin series for $h(y)$ up to and including the $y^3$ term.

[4]

Hence, find an approximate value for ${\int_{0}^{1} e^{y^2} \sin(y^2) \, dy}$ .

[4]

Hence, or otherwise, determine the value of ${\lim_{y \to 0} \frac{e^y \cos y - 1 - y}{y^3}}$ .

[3]