Question

Evaluate:
$\rm \displaystyle \lim_{x\to \pi/6}\ \frac{2- cosec \ x}{\cot^{2} x-3}$

Answer 1

Solution :

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Simplification :

( 2 - cosecx )/( cot² x - 3 )

= ( 2-cosecx )/(cosec²x-1-3)

= (2-cosecx)/[ cosec²x - 2² ]

= [-(cosecx-2)]/[(cosecx-2)(cosecx+2)]

After cancellation, we get

= -1/(cosecx+2) ---( 1 )
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Now ,

$\rm \displaystyle \lim_{x\to \pi/6}\ \frac{2- cosec \ x}{\cot^{2} x-3}$

= $\rm \displaystyle \lim_{x\to \pi/6}\ \frac{-1}{(cosec \x + 2)}$

= $\rm \displaystyle \frac{-1}{cosec \frac{π}{6}+2}$

= $\rm \displaystyle \frac{-1}{(2+2)}$

= -1/4

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