Limits of Trigonometric Functions
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
\[lim_{\theta \rightarrow 0^{{}-{}}} \frac{sin \theta}{ \theta} = lim_{\theta \rightarrow 0^{{}+{}}} \frac{sin \theta}{ \theta} = 1\]
\[lim_{\theta \rightarrow 0^{{}-{}}} \frac{tan \theta}{ \theta} = lim_{\theta \rightarrow 0^{{}+{}}} \frac{tan \theta}{ \theta} = 1\]
\[lim_{\theta \rightarrow 0} sin \theta = lim_{\theta \rightarrow 0} tan \theta = lim_{\theta \rightarrow 0} \theta\]
