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Evaluate the following limit: (lim)(x...

Evaluate the following limit: `(lim)_(x->1/(sqrt(2)))(cos^(-1)(2xsqrt(1-x^2)))/(x-1/(sqrt(2)))`

A

`sqrt(2)`

B

`-2sqrt(2)`

C

`2sqrt(2)`

D

doesn't exist

Text Solution

Verified by Experts

The correct Answer is:
C
Let `x=costheta`
`xto1/(sqrt(2)0+,thetato(pi)/4`
As `lim_(0to(pi^(-))/4)(cos^(-1)cos2theta)/(costheta-1/(sqrt(2)))=lim_(theta to (pi^(-))/4) ((pi)/2-2theta)/(costheta-1/(sqrt(2))`
`lim_(thetato(pi^(-))/4)(-2)/(-sintheta)=2sqrt(2)`
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