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lim(x->alpha)(tanxcotalpha)^(1/(x-alpha)...

`lim_(x->alpha)(tanxcotalpha)^(1/(x-alpha))` is equal to

A

`2cosec 2alpha`

B

`(1)/(2)sin 2alpha`

C

`-2cosec 2alpha`

D

none of these

Text Solution

Verified by Experts

The correct Answer is:
D
We have ,
` lim_(xtoalpha)(tanxcotalpha)^((1)/(x-a))`
`=lim_(xtoalpha){1+(tanxcotalpha-1)}((1)/(x-alpha))`
` =e^(lim_(xtoalpha)(tanxcotalpha-1)/(x-alpha))=e^(lim_(xtoalpha)(tanx-tanalpha)/(x-alpha)xx(1)/(tanalpha)`
` =e^(sec^(alpha//tanalpha=e^(2cosec 2 alpha))`.
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