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

1FoundationSAQ-SDefinition and calculation of limits5 marksPaper 1~8 min
A biologist is studying the population of a bacterial culture. The number of bacteria, N(t)N(t), after tt hours is modelled by N(t)=5000t+2000t+2,t≥0.N(t) = \frac{5000t + 2000}{t + 2}, \quad t \geq 0.
(a)
State the initial number of bacteria in the culture. [1 mark]
(b)
Find lim⁡t→∞N(t)\displaystyle\lim_{t \to \infty} N(t). [2 marks]
(c)
Explain what the value found in part (b) represents in the context of this model, and justify whether the population ever actually reaches this value. [2 marks]
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Solutions

2MasterySAQ-SDefinition and calculation of limits5 marksPaper 1~8 min
A function is defined as f(x)=x2−5x+6x−2f(x) = \dfrac{x^2 - 5x + 6}{x - 2} for x≠2x \neq 2.
(a)
Show that lim⁡x→2f(x)=−1\displaystyle\lim_{x \to 2} f(x) = -1. [3 marks]
(b)
Given that g(x)=[f(x)]2−1x−2g(x) = \dfrac{[f(x)]^2 - 1}{x - 2}, determine lim⁡x→2g(x)\displaystyle\lim_{x \to 2} g(x). [2 marks]
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Solutions

3ChallengeSAQ-LDefinition and calculation of limits8 marksPaper 1~12 min
An environmental scientist models the flow rate F(x)F(x) (thousands of litres per hour) through a filtration membrane of thickness xx mm by: F(x)=3x2−5x+2x−1,x≠1F(x) = \frac{3x^2 - 5x + 2}{x - 1}, \quad x \neq 1
(a)
Determine lim⁡x→1F(x)\displaystyle\lim_{x \to 1} F(x), showing full algebraic justification. [3 marks]
(b)
The scientist defines F(1)=1F(1) = 1 to extend FF to all xx. Using the formal definition of continuity, verify that the extended function is continuous at x=1x = 1. [1 mark]
(c)
Using the limit definition of the derivative, F′(1)=lim⁡h→0F(1+h)−F(1)hF'(1) = \lim_{h \to 0} \frac{F(1+h) - F(1)}{h} determine whether the extended function is differentiable at x=1x = 1. Justify your answer fully. [4 marks]
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Solutions

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