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Limits and Continuity · Paper 1 — Free Maths AA HL Practice Questions

What this topic covers

Calculus is where AA HL stops rewarding memorised algebra and starts rewarding process. Every mark scheme in this topic is built around specific words — show that, hence, determine, justify — and examiners are trained to withhold marks the instant your working skips a step, even when your final answer is correct. The four subtopics here build on each other in a strict chain: limits define continuity, continuity is the gatekeeper for differentiability, derivatives feed every application (tangents, optimisation, kinematics), and integration — the reverse process — feeds differential equations. Learn it in that order and the HL-only material (implicit differentiation, integration by parts, volumes of revolution, and the whole of differential equations) stops looking like extra content and starts looking like the natural next question.

Read the full Calculus lesson notes →

1FoundationSAQ-SDefinition and calculation of limits5 marksPaper 1~8 min

The function

g

is defined as

g(x) = \begin{cases} 3x + 2, & x < 1 \\ k, & x = 1 \\ 2x + 3, & x > 1 \end{cases}

where

k

is a constant.

(a)

Find

\lim_{x \to 1^-} g(x)

. [1 mark]

(b)

Find

\lim_{x \to 1^+} g(x)

. [1 mark]

(c)

Determine the value of

k

for which

g

is continuous at

x = 1

. Justify your answer. [2 marks]

(d)

For a different value

k = 7

, identify which condition(s) of continuity fail at

x = 1

and explain why. [1 mark]

Solutions

2MasterySAQ-SDefinition and calculation of limits5 marksPaper 1~8 min

Consider the function

f(x) = \dfrac{x^2 - 4}{x - 2}

, for

x \neq 2

(a)

Show that

\displaystyle\lim_{x \to 2} f(x) = 4

. [3 marks]

(b)

Hence find the value of

\displaystyle\lim_{x \to 2} \frac{[f(x)]^2 - 16}{x - 2}

. [2 marks]

Solutions

3ChallengeSAQ-LDefinition and calculation of limits8 marksPaper 1~12 min

A function is defined by

g(x) = \begin{cases} \dfrac{x^2 - 9}{x-3}, & x \neq 3 \\[6pt] k, & x = 3 \end{cases}

where

k

is a constant.

(a)

Determine

\lim_{x \to 3} g(x)

by simplifying the expression for

x \neq 3

. [2 marks]

(b)

Determine the value of

k

for which

g

is continuous at

x = 3

. [1 mark]

(c)

For

k = 0

, classify the discontinuity of

g

x = 3

as removable or non-removable. Justify your answer using the formal definition of continuity. [5 marks]

Solutions

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