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Math Help - Fantastic limit

  1. #1
    Super Member dhiab's Avatar
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    Fantastic limit

    a real paramtre and

    Calculate :
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  2. #2
    Senior Member pankaj's Avatar
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    Quote Originally Posted by dhiab View Post
    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
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  3. #3
    Super Member dhiab's Avatar
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    Quote Originally Posted by pankaj View Post
    \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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