Problem Set 1.4 : Limit Involving Trigonometric Functions

$ 7.  \lim_{\theta \to 0} \frac{sin\, 3\theta}{tan \, \theta} $

$= \lim_{\theta \to 0} \frac{sin \, 3\theta}{\frac{sin \theta}{cos \theta}} $

$ = \lim_{\theta \to 0} \frac{cos\theta \, sin 3\theta}{sin \theta} $

$ = \lim_{\theta \to 0} [ \frac{sin 3\theta}{3 \theta}. 3\, cos \theta . \frac{1}{sin \theta} . \theta ] $ 

$ = 3 \lim_{\theta \to 0} [ \frac{sin 3\theta}{3\theta}. cos \theta. \frac{\theta}{sin \theta}] $

$ = 3 (1.1.1)~=~3 $

$ 8.  \lim_{\theta \to 0} \frac{tan 5\theta}{sin 2\theta} $

$ = \lim_{\theta \to 0} \frac{\frac{sin 5\theta}{cos 5\theta}}{sin 2\theta} $

$ = lim_{\theta \to 0} \frac{sin5 \theta}{ cos 5\theta \, sin 2\theta} $

$ = lim_{\theta \to 0} [\frac{sin 5\theta}{5\theta}.\frac{2\theta}{sin 2\theta}.\frac{1}{cos 5\theta}.5\theta. \frac{1}{2\theta}] $

$ = lim_{\theta \to 0} [\frac{sin 5\theta}{5\theta}.\frac{2\theta}{sin 2\theta}.\frac{1}{cos 5\theta}.\frac{5}{2}]  $

$ =  \frac{5}{2} lim_{\theta \to 0} [\frac{sin 5\theta}{5\theta}.\frac{2\theta}{sin 2\theta}.\frac{1}{cos 5\theta}] $

$ = \frac{5}{2} (1.1.1)~=~ \frac{5}{2} $

 

$9. \lim_{\theta \to 0} \frac{cot\, \pi \theta \, sin \theta}{2 sec \theta}$

$ = \lim_{\theta \to 0} \frac{\frac{cos \pi \theta}{sin \pi \theta} sin \theta}{\frac{2}{cos \theta}}$

$ = \lim_{\theta \to 0} \frac{cos \pi \theta \, sin \theta \, cos \theta}{2 sin \pi \theta} $

$= \lim_{\theta \to 0} [\frac{sin \theta}{\theta}. \frac{\pi \theta}{sin \pi \theta}. \frac{1}{\pi}. \frac{cos \pi \theta \, cos \theta}{2}]$

$ = \frac{1}{2\pi} ~\lim_{\theta \to 0}  [\frac{sin \theta}{\theta}. \frac{\pi \theta}{sin \pi \theta}. cos \pi \theta \, cos \theta ]$

$ =\frac{1}{2 \pi}(1.1.1) ~=~\frac{1}{2 \pi}$