QUESTION 1

QUESTION 1

Using l’Hôpital’s Rule, determine the value of $$\mathop {\lim }\limits_{x \to 0} \frac{{\tan x - x}}{{1 - \cos x}} .$$

MARKSCHEME 1

MARKSCHEME 1

$\mathop {\lim }\limits_{x \to 0} \frac{{\tan x - x}}{{1 - \cos x}} = \mathop {\lim }\limits_{x \to 0} \frac{{{{\sec }^2}x - 1}}{{\sin x}}$     M1A1A1

this still gives $\frac{0}{0}$

EITHER

repeat the process     M1

$= \mathop {\lim }\limits_{x \to 0} \frac{{2{{\sec }^2}x\tan x}}{{\cos x}}$     A1

$= 0$     A1

OR

$= \mathop {\lim }\limits_{x \to 0} \frac{{{{\tan }^2}x}}{{\sin x}}$     M1

$= \mathop {\lim }\limits_{x \to 0} \frac{{\sin x}}{{{{\cos }^2}x}}$     A1

$= 0$     A1

[6 marks]