Leibnitz's theorem and L'Hospital's ruleEdexcel A-Level Further Maths: Subtopic test

1. 1
   Let $y=x^2\mathrm{e}^{3x}$.

   (a)

   Use Leibnitz's theorem to find $\dfrac{\mathrm{d}^2y}{\mathrm{d}x^2}$.

   [1 mark]

   • A$\mathrm{e}^{3x}(9x^2+2)$
   • B$\mathrm{e}^{3x}(9x^2+6x+2)$
   • C$\mathrm{e}^{3x}(3x^2+12x+2)$
   • D$\mathrm{e}^{3x}(9x^2+12x+2)$

   (b)

   Find the value of $\dfrac{\mathrm{d}^3y}{\mathrm{d}x^3}$ at $x=0$.

   [1 mark]

   • A$18$
   • B$6$
   • C$54$
   • D$0$

   (c)

   Use Leibnitz's theorem to find the value of $\dfrac{\mathrm{d}^4y}{\mathrm{d}x^4}$ at $x=0$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   L'Hospital's rule states that if $f(x)$ and $g(x)$ both tend to $0$, or both tend to $\pm\infty$, as $x\to a$, then $\displaystyle\lim_{x\to a}\frac{f(x)}{g(x)}=\lim_{x\to a}\frac{f'(x)}{g'(x)}$, provided the second limit exists.

   (a)

   Find $\displaystyle\lim_{x\to0}\frac{\sin3x}{\sin5x}$.

   [1 mark]

   • A$\frac53$
   • B$1$
   • C$\frac35$
   • D$0$

   (b)

   Find $\displaystyle\lim_{x\to\infty}\frac{\mathrm{e}^x}{x^2}$.

   [1 mark]

   • A$0$
   • B$\infty$
   • C$\frac12$
   • D$1$

   (c)

   A student evaluates $\displaystyle\lim_{x\to0}\frac{1+\cos x}{x^2}$ as follows: “This is $\frac00$. Differentiating gives $\frac{-\sin x}{2x}$, which is again $\frac00$. Differentiating again gives $\frac{-\cos x}{2}\to-\frac12$.” Explain the error in the student's working and describe the true behaviour of the quotient as $x\to0$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Let $f(x)=\dfrac{2\sin x-\sin2x}{x-\sin x}$ for $x\neq0$.

   (a)

   Show that applying L'Hospital's rule once to $\displaystyle\lim_{x\to0}f(x)$ gives an expression that is again of the form $\frac00$.

   [3 marks]

   (b)

   Hence find $\displaystyle\lim_{x\to0}f(x)$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Let $a$ be a positive constant.

   (a)

   Show that $\displaystyle\lim_{x\to\infty}\left(1+\frac{a}{x}\right)^{x}=\mathrm{e}^{a}$.

   [6 marks]

   (b)

   (i) Use the result in (a) to find $\displaystyle\lim_{x\to\infty}\left(1+\frac3x\right)^{2x}$.
   (ii) Find $\displaystyle\lim_{x\to\infty}x\left(\mathrm{e}^{a/x}-1\right)$.

   [6 marks]

   Total for question 4: 12 marks

End of questions