Question7.2a - Derivatives of the Sine an
1. Evaluate $\lim _{\theta \rightarrow 0} \frac{\sin 4 \theta}{\sin 6 \theta}=\frac{2}{3}$
2. Evaluate $\lim _{\theta \rightarrow 0} \frac{\sin \theta}{\theta+\tan \theta}$
3. Evaluate $\lim _{x \rightarrow 1} \frac{\sin (x-1)}{x^{2}+x-2}$

Studdy Solution
Apply L'Hôpital's Rule since both the numerator and denominator approach 0:
Differentiate the numerator and denominator:
$= \lim _{x \rightarrow 1} \frac{\cos (x-1)}{2x + 1}$
Evaluate the limit:
$= \frac{\cos(0)}{2(1) + 1} = \frac{1}{3}$
The evaluated limits are:
1. $\frac{2}{3}$
2. $1$
3. $\frac{1}{3}$