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If lim(xtoa)f(x)=1 and lim(xtoa)g(x)=oo ...

If `lim_(xtoa)f(x)=1` and `lim_(xtoa)g(x)=oo` then `lim_(xtoa){f(x)}^(g(x))=e^(lim_(xtoa)(f(x)-1)xg(x))`
`lim_(xto0)((x-1+cosx)/x)^(1/x)` is equal to

A

e

B

`sqrte`

C

`(1)/(sqrte)`

D

`e^(2)`

Text Solution

Verified by Experts

The correct Answer is:
C
`lim_(x to 0) ((x-1+cosx)/(x))^(1//x) (1^(oo)), " "e^(lim_(x to o))((x-1+cos x)/(x)-1)(1)/(x)=e^(lim_(x to0) ((1-cosx))/(x^(2)))=e^(-1//2)`
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