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