## Evaluating Limits of Trigonometric Functions

To evaluate the limit of a function such as
$\frac{sin^2 \theta sin 4 \theta}{\theta^3}$
as
$\theta rightarrow 0$
we can use the following limits. For example, as
$\theta$
approaches
$\pi /2$
from below,(written
$lim_{\theta \rightarrow {\frac{\pi}{2}}^{{}-{}}}$
)
$tan \theta \rightarrow \infty$
.
We write
$lim_{\theta \rightarrow {\frac{\pi}{2}}^{{}-{}}} tan \theta = \infty$

Similarly
$lim_{\theta \rightarrow {\frac{\pi}{2}}^{{}+{}}} tan \theta = - \infty$

$lim_{\theta \rightarrow {\frac{\pi}{2}}^{{}-{}}} sec \theta = lim_{\theta \rightarrow {\frac{\pi}{2}}^{{}-{}}} \frac{1}{cos \theta} = \infty$

$lim_{\theta \rightarrow 0} sin \theta = 0$

$lim_{\theta \rightarrow 0} cos \theta = 1$

$lim_{\theta \rightarrow {\frac{\pi}{2}}^{{}+{}}} sec \theta = lim_{\theta \rightarrow {\frac{\pi}{2}}^{{}+{}}} \frac{1}{cos \theta}= - \infty$

$lim_{\theta \rightarrow {\frac{\pi}{2}}^{{}+{}}} cosec \theta = lim_{\theta \rightarrow 0^{{}+{}}} \frac{1}{sin \theta} = - \infty$

$lim_{\theta \rightarrow 0^{{}+{}}} cosec \theta = lim_{\theta \rightarrow 0^{{}+{}}} \frac{1}{sin \theta} = \infty$

Three very useful limits are
$lim_{\theta \rightarrow 0^{{}-{}}} \frac{sin \theta}{ \theta} = lim_{\theta \rightarrow 0^{{}+{}}} \frac{sin \theta}{ \theta} = 1$

$lim_{\theta \rightarrow 0^{{}-{}}} \frac{tan \theta}{ \theta} = lim_{\theta \rightarrow 0^{{}+{}}} \frac{tan \theta}{ \theta} = 1$

$lim_{\theta \rightarrow 0} sin \theta = lim_{\theta \rightarrow 0} tan \theta = lim_{\theta \rightarrow 0} \theta$

Hence
$lim_{\theta \rightarrow 0} \frac{sin^2 \theta sin 4 \theta}{\theta^3} =\frac{\theta^2 \times 4 \theta}{\theta^3} = 4$

$lim_{n \rightarrow \infty} n sin(\frac{2 \pi}{n})=n \times \frac{2 \pi}{m}=2 \pi$