1. 1
   Let $f(x) = e^{3x} - 1$ and $g(x) = \sin 2x$, for $x \in \mathbb{R}$.

   (a)

   Find $\lim_{x \to 0}\dfrac{f(x)}{g(x)}$.

   [1 mark]

   • A$\dfrac{3}{2}$
   • B$\dfrac{2}{3}$
   • C$3$
   • D$0$

   (b)

   Find $\lim_{x \to 0}\dfrac{f(x) - 3x}{x^{2}}$.

   [1 mark]

   • A$9$
   • B$\dfrac{3}{2}$
   • C$\dfrac{9}{2}$
   • DThe limit does not exist.

   (c)

   Find $\lim_{x \to 0}\dfrac{g(x)}{f(2x)}$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Consider the functions defined for $x > 0$ by $u(x) = x^{2}e^{-x}$, $v(x) = \dfrac{5x^{2} - 3x}{2x^{2} + \ln x}$ and $w(x) = x\ln x$.

   (a)

   Find $\lim_{x \to \infty} u(x)$.

   [1 mark]

   • A$2$
   • B$1$
   • C$\infty$
   • D$0$

   (b)

   Find $\lim_{x \to \infty} v(x)$.

   [1 mark]

   • A$0$
   • B$\dfrac{5}{2}$
   • C$\infty$
   • D$5$

   (c)

   By writing $w(x)$ as a quotient, find $\lim_{x \to 0^{+}} w(x)$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A bank pays a nominal annual interest rate of 6%, compounded $n$ times per year. An investment of £1000 grows to £$A_{n}$ after one year, where $A_{n} = 1000\left(1 + \dfrac{0.06}{n}\right)^{n}$. In this question $n$ may be treated as a continuous variable.

   (a)

   Show that $\lim_{n \to \infty} n\ln\left(1 + \dfrac{0.06}{n}\right) = 0.06$.

   [3 marks]

   (b)

   Hence find, in exact form, the limiting value of $A_{n}$ as $n \to \infty$, and interpret this value in context. Use your calculator to find by how much this limiting value exceeds the amount after one year when interest is compounded once a year.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   This question is about limits as $x \to 0$ in which the numerator and the denominator both tend to $0$. The Maclaurin series for $e^{x}$ is $1 + x + \dfrac{x^{2}}{2!} + \dfrac{x^{3}}{3!} + \cdots$.

   (a)

   Use l'Hôpital's rule to find $\lim_{x \to 0}\dfrac{x - \sin x}{x^{3}}$, showing clearly that the rule may be applied at each stage.

   [6 marks]

   (b)

   It is given that $\lim_{x \to 0}\dfrac{e^{ax} - 1 - 2x}{x^{2}}$ exists and is finite, where $a$ is a constant. Use the Maclaurin series for $e^{x}$ to find the value of $a$ and the value of the limit.

   [6 marks]

   Total for question 4: 12 marks

End of questions