First slide

Introduction to limits

Question

Evaluate $\lim_{n \to \infty} {\frac{1}{2} \tan \frac{x}{2} + \frac{1}{2^{2}} \tan \frac{x}{2^{2}} + .... + \frac{1}{2^{n}} \tan \frac{x}{2^{n}}}$

Moderate

A

$x \tan \frac{x}{2}$
B

$\frac{1}{x} \cot \frac{x}{2}$
C

$\frac{x - \cot x}{2}$
D

$\frac{1}{x} - \cot x$

Solution

$\lim_{n \to \infty} {\frac{1}{2} \tan \frac{x}{2} + \frac{1}{2^{2}} \tan \frac{x}{2^{2}} + .... + \frac{1}{2^{n}} \tan \frac{x}{2^{n}}}$ = $\lim_{n \to \infty} {- \cot x + (\cot x + \frac{1}{2} \tan \frac{x}{2} + \frac{1}{2^{2}} \tan \frac{x}{2^{2}} + .... + \frac{1}{2^{n}} \tan \frac{x}{2^{n}})}$ $\lim_{n \to \infty} {- \cot x + (\frac{1}{2} \cot \frac{x}{2} + \frac{1}{2^{2}} \tan \frac{x}{2^{2}}) + .... + \frac{1}{2^{n}} \tan \frac{x}{2^{n}}}$ $(∵ \cot x + \frac{1}{2} \tan \frac{x}{2} = \frac{1}{2} \cot \frac{x}{2})$ Proceeding like this we get $= \lim_{n \to \infty} {- \cot x + \frac{1}{2^{n}} \cot \frac{x}{2^{n}}}$ $= - \cot x + \lim_{n \to \infty} (\frac{x / 2^{n}}{\tan \frac{x}{2^{n}}} . \frac{1}{x}) = - \cot x + \frac{1}{x}$

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