# Evaluation of Trigonometric Limits – Tan

## Evaluation of Trigonometric Limits – Tan

$$\underset{\theta \to 0}{\mathop{Lim}}\,\frac{\tan \theta }{\theta }=1$$,

Proof:

$$\underset{\theta \to 0}{\mathop{Lim}}\,\frac{\tan \theta }{\theta }=1$$,

$$\underset{\theta \to 0}{\mathop{Lim}}\,\frac{\tan \theta }{\theta }=\underset{\theta \to 0}{\mathop{Lim}}\,\frac{\sin \theta }{\theta }\times \frac{1}{\cos \theta }$$,

$$=\underset{\theta \to 0}{\mathop{Lim}}\,\frac{\sin \theta }{\theta }\times \underset{\theta \to 0}{\mathop{Lim}}\,\frac{1}{\cos \theta }$$,

$$\underset{\theta \to 0}{\mathop{\because Lim}}\,\frac{\sin \theta }{\theta }=1$$,

$$1\times \underset{\theta \to 0}{\mathop{Lim}}\,\frac{1}{\cos \theta }$$,

$$1\times 1=1$$,

$$\underset{\theta \to 0}{\mathop{Lim}}\,\frac{\tan \theta }{\theta }=1$$,

Hence proved

Example: Evaluate $$\underset{\theta \to \infty }{\mathop{Lim}}\,{{2}^{x-1}}\tan \left( \frac{a}{{{2}^{x}}} \right)$$,

Solution:

$$\underset{\theta \to \infty }{\mathop{Lim}}\,{{2}^{x-1}}\tan \left( \frac{a}{{{2}^{x}}} \right)$$,

$$\underset{\theta \to \infty }{\mathop{Lim}}\,{{2}^{x}}\times {{2}^{-1}}\tan \left( \frac{a}{{{2}^{x}}} \right)$$,

$$\underset{\theta \to \infty }{\mathop{Lim}}\,{{2}^{x}}\times \frac{1}{2}\tan \left( \frac{a}{{{2}^{x}}} \right)$$,

$$\underset{\theta \to \infty }{\mathop{Lim}}\,\frac{a}{2}\frac{\tan \left( \frac{a}{{{2}^{x}}} \right)}{\frac{a}{{{2}^{x}}}}$$,

Let us consider $$y=\frac{a}{{{2}^{x}}}$$,

$$\underset{\theta \to o}{\mathop{Lim}}\,\frac{a}{2}\frac{\tan y}{y}$$ ,

$$\because \underset{\theta \to 0}{\mathop{Lim}}\,\frac{\tan \theta }{\theta }=1$$,

= a/2