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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  
\[lim_{\theta \rightarrow 0} \frac{sin^2 \theta sin 4 \theta}{\theta^3} =\frac{\theta^2 \times 4 \theta}{\theta^3} = 4\]

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

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