Leibnitz's theorem and L'Hospital's ruleEdexcel A-Level Further Maths: Mind map

Exam questions on Leibnitz's theorem and L'Hospital's rule

1. Let $y=x^2\mathrm{e}^{3x}$.
   Use Leibnitz's theorem to find the value of $\dfrac{\mathrm{d}^4y}{\mathrm{d}x^4}$ at $x=0$.2 marks
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 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
3. Let $f(x)=\dfrac{2\sin x-\sin2x}{x-\sin x}$ for $x\neq0$.
   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