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