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

State whether $\lim_{x \to 2} g(x)$ exists, justifying your answer. [1 mark]

Determine whether $g$ is continuous at $x = 2$. Justify your answer fully. [2 marks]

Hence, state the value of $f(1)$ that would make $f$ continuous at $x = 1$. [1 mark]

The function $g$ is defined as $$g(x) = \begin{cases} \dfrac{x^2 - 1}{x - 1}, & x \neq 1 \\[6pt] k, & x = 1 \end{cases}$$ Determine the value of $k$ for which $g$ is continuous at $x = 1$, and state whether $g(x) = x + 1$ for all $x \in \mathbb{R}$. Justify your answer. [2 marks]

Define $g(x) = \dfrac{x^2 - 4}{x - 2} - 4$ for $x \neq 2$. Show that $g(x) = x - 2$, and hence calculate $\displaystyle\lim_{x \to 2} \dfrac{g(x)}{x - 2}$. [2 marks]

The result in (b) is the limit of the difference quotient for $f$ as $x \to 2$ — the same limit that would define $f'(2)$ if $f$ were defined at $x = 2$. Explain why this is the case, and state the value of this limit. [1 mark]

Hence determine $\lim_{x \to 2^+} g(x)$ and $\lim_{x \to 2^-} g(x)$, and state whether $g$ has a removable discontinuity at $x = 2$. Justify your answer. [3 marks]

Using your results from (a), write down the equations of the two lines that the graph of $g$ coincides with, away from $x = 2$. Hence sketch the graph of $g$, clearly labelling these lines, the open circles at $x = 2$, and the $y$-intercept. [3 marks]