finding limits as x tends to pi/4

**alexandrabel90** · January 8th 2010, 09:11 PM

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

thanks!

**Prove It** · January 8th 2010, 09:24 PM

Originally Posted by alexandrabel90

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

thanks!

Remember that $\sin^2{x} + \cos^2{x} = 1$

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

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

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

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

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

$= \cot{x} + 1$ .

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

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

$= 1 + 1$

$= 2$ .

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