Limits using Maclaurin series and L'Hopital's ruleAQA A-Level Further Maths: Subtopic test

1. 1
   Let $L=\lim_{x\to0}\dfrac{1-\cos2x}{x^2}$.

   (a)

   What happens when $x=0$ is substituted directly into $\frac{1-\cos2x}{x^2}$?

   [1 mark]

   • Athe value is $0$
   • Bthe value is $1$
   • Cthe value is $\infty$
   • Dit gives the indeterminate form $\frac00$

   (b)

   Which expression results from one application of l'Hôpital's rule?

   [1 mark]

   • A$\frac{2\sin2x}{2x}$
   • B$\frac{\sin2x}{2x}$
   • C$\frac{-2\sin2x}{2x}$
   • D$\frac{2\cos2x}{2x}$

   (c)

   Use the Maclaurin series for $\cos x$ to find the value of $L$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Let $M=\lim_{x\to\infty}\dfrac{x^2}{\mathrm{e}^{x}}$.

   (a)

   What is the form of $\frac{x^2}{\mathrm{e}^x}$ as $x\to\infty$?

   [1 mark]

   • A$\frac00$
   • B$\frac{\infty}{\infty}$
   • C$0\times\infty$
   • D$\infty-\infty$

   (b)

   What is the value of $M$?

   [1 mark]

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

   (c)

   Show how applying l'Hôpital's rule twice gives your answer to (b).

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Let $F(x)=\dfrac{\sin x-x\cos x}{x^3}$ for $x\ne0$.

   (a)

   Use Maclaurin series to find $\lim_{x\to0}F(x)$.

   [3 marks]

   (b)

   Verify your answer to (a) using l'Hôpital's rule.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Let $g(x)=\dfrac{\mathrm{e}^{x}-1-x}{x^2}$ for $x\ne0$.

   (a)

   (i) Use the Maclaurin series for $\mathrm{e}^x$ to find $\lim_{x\to0}g(x)$.
   (ii) Confirm your answer using l'Hôpital's rule.

   [6 marks]

   (b)

   Find the value of $\lim_{x\to0}\dfrac{g(x)-\frac12}{x}$.

   [6 marks]

   Total for question 4: 12 marks

End of questions