## Evaluating Limits of Trigonometric Functions

To evaluate the limit of a function such as  {jatex options:inline}\frac{sin^2 \theta sin 4 \theta}{\theta^3}{/jatex}  as  {jatex options:inline}\theta rightarrow 0{/jatex}  we can use the following limits. For example, as  {jatex options:inline}\theta{/jatex}  approaches  {jatex options:inline}\pi /2{/jatex}  from below,(written  {jatex options:inline}lim_{\theta \rightarrow {\frac{\pi}{2}}^{{}-{}}}{/jatex})  {jatex options:inline}tan \theta \rightarrow \infty{/jatex}.
We write  {jatex options:inline}lim_{\theta \rightarrow {\frac{\pi}{2}}^{{}-{}}} tan \theta = \infty{/jatex}
Similarly  {jatex options:inline}lim_{\theta \rightarrow {\frac{\pi}{2}}^{{}+{}}} tan \theta = - \infty{/jatex}
{jatex options:inline}lim_{\theta \rightarrow {\frac{\pi}{2}}^{{}-{}}} sec \theta = lim_{\theta \rightarrow {\frac{\pi}{2}}^{{}-{}}} \frac{1}{cos \theta} = \infty{/jatex}
{jatex options:inline}lim_{\theta \rightarrow 0} sin \theta = 0{/jatex}
{jatex options:inline}lim_{\theta \rightarrow 0} cos \theta = 1{/jatex}
{jatex options:inline}lim_{\theta \rightarrow {\frac{\pi}{2}}^{{}+{}}} sec \theta = lim_{\theta \rightarrow {\frac{\pi}{2}}^{{}+{}}} \frac{1}{cos \theta}= - \infty{/jatex}
{jatex options:inline}lim_{\theta \rightarrow {\frac{\pi}{2}}^{{}+{}}} cosec \theta = lim_{\theta \rightarrow 0^{{}+{}}} \frac{1}{sin \theta} = - \infty{/jatex}
{jatex options:inline}lim_{\theta \rightarrow 0^{{}+{}}} cosec \theta = lim_{\theta \rightarrow 0^{{}+{}}} \frac{1}{sin \theta} = \infty{/jatex}
Three very useful limits are
{jatex options:inline}lim_{\theta \rightarrow 0^{{}-{}}} \frac{sin \theta}{ \theta} = lim_{\theta \rightarrow 0^{{}+{}}} \frac{sin \theta}{ \theta} = 1{/jatex}
{jatex options:inline}lim_{\theta \rightarrow 0^{{}-{}}} \frac{tan \theta}{ \theta} = lim_{\theta \rightarrow 0^{{}+{}}} \frac{tan \theta}{ \theta} = 1{/jatex}
{jatex options:inline}lim_{\theta \rightarrow 0} sin \theta = lim_{\theta \rightarrow 0} tan \theta = lim_{\theta \rightarrow 0} \theta{/jatex}
Hence  {jatex options:inline}lim_{\theta \rightarrow 0} \frac{sin^2 \theta sin 4 \theta}{\theta^3} =\frac{\theta^2 \times 4 \theta}{\theta^3} = 4{/jatex}
{jatex options:inline}lim_{n \rightarrow \infty} n sin(\frac{2 \pi}{n})=n \times \frac{2 \pi}{m}=2 \pi {/jatex}