You're viewing 1 free preview question for Maths AA HL — subscribe to unlock 16 more, plus the Full Question Bank, Mock Papers & Worksheets.Upgrade

Limits and Continuity · Paper 3 — 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 →

1ChallengeLAQDefinition and calculation of limits10 marksPaper 3~15 min

A function

h

is defined by

h(x) = \dfrac{\sqrt{x^2 + 2x + 5} - (x+1)}{x}

for

x \neq 0

(a)

Show that

h(x)

can be written in the form

h(x) = \sqrt{1 + \dfrac{2}{x} + \dfrac{5}{x^2}} - 1 - \dfrac{1}{x}

for

x > 0

, and hence prove that

\displaystyle\lim_{x \to \infty} h(x) = 0

. [4 marks]

(b)

Show that for

x < 0

h(x) = -\sqrt{1 + \frac{2}{x} + \frac{5}{x^2}} - 1 - \frac{1}{x}.

Hence determine

\displaystyle\lim_{x \to -\infty} h(x)

. [3 marks]

(c)

The graph of

y = h(x)

has two distinct horizontal asymptotes. State both asymptotes and explain why the asymptote as

x \to -\infty

differs from the asymptote as

x \to \infty

, with reference to the role of

\sqrt{x^2} = |x|

in the simplification of

h(x)

. [3 marks]

Solutions

Questions 2-17

Unlock these questions with Premium

That's every free question here — 1 of 17.

+16 more questions, plus Mock Papers & Worksheets, are waiting in Premium.

Upgrade