Functions, limits and continuity – university Year 1 maths test to download as PDF

1. Evaluate \(\displaystyle\lim_{x\to3}\dfrac{x^2-9}{x-3}\). (2 marks)
2. Evaluate \(\displaystyle\lim_{x\to0}\dfrac{\sin2x}{\sin3x}\). (2 marks)

Prove from the definition that \(\displaystyle\lim_{x\to4}(2x+1)=9\).

Let \(f(x)=\begin{cases}x^2+a&x<1\\ 3x-2&x\ge1\end{cases}\).

1. Find the value of \(a\) for which \(f\) is continuous at \(x=1\). (3 marks)
2. Is \(f\) then differentiable at 1? (1 mark)

A shelf of length \(x\) metres must satisfy \(f(x)=x^3+2x-6=0\) for a design.

1. Show that there is a root between 1 and 2. (2 marks)
2. Use one bisection step to find a smaller interval. (2 marks)

Show that \(\displaystyle\lim_{x\to0}x^2\cos\dfrac1x=0\), and explain why \(\displaystyle\lim_{x\to0}\cos\dfrac1x\) does not exist.