Limit using l'Hôpital's rule

A question is this type if and only if it explicitly asks to use l'Hôpital's rule to evaluate a limit (may be as x→0, x→∞, or x→other values).

4 questions · Standard +0.2

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Edexcel FP1 2020 June Q1
5 marks Standard +0.3
  1. Use l'Hospital's Rule to show that
$$\lim _ { x \rightarrow \frac { \pi } { 2 } } \frac { \left( e ^ { \sin x } - \cos ( 3 x ) - e \right) } { \tan ( 2 x ) } = - \frac { 3 } { 2 }$$
Edexcel FP1 2024 June Q3
6 marks Standard +0.3
  1. Use L'Hospital's rule to show that
$$\lim _ { x \rightarrow 0 } \left( \frac { 1 } { \sin x } - \frac { 1 } { x } \right) = 0$$ (6)
AQA Further Paper 1 2021 June Q9
4 marks Moderate -0.5
Use l'Hôpital's rule to show that $$\lim_{x \to \infty} (xe^{-x}) = 0$$ Fully justify your answer. [4 marks]
AQA Further Paper 1 2023 June Q13
5 marks Standard +0.8
Use l'Hôpital's rule to prove that $$\lim_{x \to \pi} \frac{x \sin 2x}{\cos\left(\frac{x}{2}\right)} = -4\pi$$ [5 marks]