how do you find the lim of ( cosec^2 x -2)/ ( cot x -1 ) as x tends to pi/4?

thanks!

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- Jan 8th 2010, 09:11 PMalexandrabel90finding limits as x tends to pi/4
how do you find the lim of ( cosec^2 x -2)/ ( cot x -1 ) as x tends to pi/4?

thanks! - Jan 8th 2010, 09:24 PMProve It
Remember that $\displaystyle \sin^2{x} + \cos^2{x} = 1$

Dividing both sides by $\displaystyle \sin^2{x}$ gives us

$\displaystyle 1 + \cot^2{x} = \csc^2{x}$.

So $\displaystyle \frac{\csc^2{x} - 2}{\cot{x} - 1} = \frac{1 + \cot^2{x} - 2}{\cot{x} - 1}$

$\displaystyle = \frac{\cot^2{x} - 1}{\cot{x} - 1}$

$\displaystyle = \frac{(\cot{x} - 1)(\cot{x} + 1)}{\cot{x} - 1}$

$\displaystyle = \cot{x} + 1$.

So $\displaystyle \lim_{x \to \frac{\pi}{4}}\frac{\csc^2{x} - 2}{\cot{x} - 1} = \lim_{x \to \frac{\pi}{4}}(\cot{x} + 1)$

$\displaystyle = \cot{\frac{\pi}{4}} + 1$

$\displaystyle = 1 + 1$

$\displaystyle = 2$.