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

1FoundationSAQ-SDefinition and calculation of limits5 marksPaper 1~8 min
Consider the function f(x)={2x+1,x<25,x=2x2−1,x>2f(x) = \begin{cases} 2x + 1, & x < 2 \\ 5, & x = 2 \\ x^2 - 1, & x > 2 \end{cases}
(a)
Find lim⁡x→2−f(x)\lim_{x \to 2^-} f(x) and lim⁡x→2+f(x)\lim_{x \to 2^+} f(x). [2 marks]
(b)
State whether lim⁡x→2f(x)\lim_{x \to 2} f(x) exists. Give a reason for your answer. [1 mark]
(c)
Determine whether ff is continuous at x=2x = 2. Justify your answer fully. [2 marks]
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Solutions

2FoundationSAQ-SDefinition and calculation of limits5 marksPaper 1~8 min
A function gg is defined by g(x)={3x−4,x<12,x=15−x,x>1g(x) = \begin{cases} 3x - 4, & x < 1 \\ 2, & x = 1 \\ 5 - x, & x > 1 \end{cases}
(a)
Calculate lim⁡x→1−g(x)\displaystyle\lim_{x \to 1^-} g(x) and lim⁡x→1+g(x)\displaystyle\lim_{x \to 1^+} g(x). [2 marks]
(b)
State whether lim⁡x→1g(x)\displaystyle\lim_{x \to 1} g(x) exists. Justify your answer. [1 mark]
(c)
Determine whether gg is continuous at x=1x = 1. Justify your answer using the values found in part (a) and the value g(1)=2g(1) = 2. [2 marks]
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Solutions

3MasterySAQ-SDefinition and calculation of limits5 marksPaper 1~8 min
A company models the cost, C(x)C(x) dollars, of producing xx units of a product by the function C(x)=1000x+5x,x>0.C(x) = \frac{1000}{x} + 5x, \quad x > 0.
(a)
Show that C(x)C(x) can be written as 1000+5x2x\dfrac{1000 + 5x^2}{x}. [1 mark]
(b)
Find C′(x)C'(x). [2 marks]
(c)
Find the value of xx that minimises C(x)C(x), and calculate the minimum cost. [2 marks]
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Solutions

4ChallengeSAQ-LDefinition and calculation of limits8 marksPaper 1~12 min
The population of a rare species of bird on an island is modelled by P(t)=500tt+5P(t) = \dfrac{500t}{t+5}, where tt is the time in years since monitoring began (t≥0t \geq 0).
(a)
Calculate P(5)P(5) and state what this value represents in context. [2 marks]
(b)
Determine lim⁡t→∞P(t)\displaystyle\lim_{t \to \infty} P(t) and interpret its meaning in context. [3 marks]
(c)
A conservationist claims the model predicts the population will eventually stabilise at exactly 500 birds. Evaluate this claim, using the behaviour of P(t)P(t) for large tt. [3 marks]
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Solutions

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