Fantastic limit

**dhiab** · June 21st 2009, 09:46 AM

a real paramtre and

Calculate :

**pankaj** · June 21st 2009, 10:20 AM

Originally Posted by dhiab

a real paramtre and

Calculate :

$\lim_{x\to a}\left( \frac{\tan x- \tan a}{x-a}-\frac{\sin x-\sin a}{x-a} \right) \frac{x-a}{x-a-(\sin x-\sin a)}$

= $\lim_{x\to a}\left( \frac{\tan x- \tan a}{x-a}-\frac{\sin x-\sin a}{x-a} \right) \frac{1}{1-\frac{\sin x-\sin a}{x-a}}$

$<br /> =\frac{\sec^2a-\cos a}{1-\cos a}<br />$

$\left(since, \lim_{x\to a}\frac{f(x)-f(a)}{x-a}=f'(a)\right)$

You may simplify further but I am bored.

Alternatively you may use the L'Hospital Rule

**dhiab** · June 21st 2009, 11:51 AM

Originally Posted by pankaj

$\lim_{x\to a}\left( \frac{\tan x- \tan a}{x-a}-\frac{\sin x-\sin a}{x-a} \right) \frac{x-a}{x-a-(\sin x-\sin a)}$

= $\lim_{x\to a}\left( \frac{\tan x- \tan a}{x-a}-\frac{\sin x-\sin a}{x-a} \right) \frac{1}{1-\frac{\sin x-\sin a}{x-a}}$

$<br /> =\frac{\sec^2a-\cos a}{1-\cos a}<br />$

$\left(since, \lim_{x\to a}\frac{f(x)-f(a)}{x-a}=f'(a)\right)$

You may simplify further but I am bored.

Alternatively you may use the L'Hospital Rule

Hello : Thank you, I'have anathor resolution with the L'Hospital Rule

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