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Let f(x)=(sin^(-1)(1-{x})xxcos^(-1)(1-{x...

`Let f(x)=(sin^(-1)(1-{x})xxcos^(-1)(1-{x}))/(sqrt(2{x})xx(1-{x}))`, where `{x}` denotes the fractional part of x.
Which of the following is true?

A

`"Lt". _(x to 0^(+))f(x)=sqrt(2) "Lt"_(xto 0^(-)) f(x)`

B

`"Lt."_(xto 0^(-))f(x)=sqrt(2) "Lt."_(x to 0^(+))f(x)`

C

`"Lt"_(x to 0^(-))f(x)=(pi)/(2sqrt(2))`

D

`"Lt"_(xto 0^(-))f(x)=pisqrt(2)`

Text Solution

Verified by Experts

The correct Answer is:
A, C
`:' xto 0^(-) implies{x} =1+x`
`:. lim(xto 0^(-)) f(x)=lim_(xto 0) (sin^(-1)(-x)cos^(-1)(-))/(sqrt(2(1+x)).(-x))=(pi)/(2sqrt(2))`
& for `xto 0^(+)implies{x}=x`
`lim_(xto 0^(+)) f(x)=lim_(xto0) (sin^(-1)(1-x)cos^(-1)(1-x))/(sqrt(2x)(1-x))=(pi)/2`
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