For what value of k, the function defined by
`f(x)={((log(1+2x)sin x^(@))/(x^(2))",", "for "x ne 0),(k",","for "x =0):}`
is continuous at x = 0 ?

Options

A. 2
B. `(1)/(2)`
C. `(pi)/(90)`
D. `(90)/(pi)`

Answer

C

Explanation

Given `p(x)={{:(,(log(1+2x)sinx^(@))/(x^(2)),"for x "ne 0),(,k, "for x=0"):}`
`={{:(,(log(1+2x)sin""(pix)/(180))/(x^(2)),"for x"ne 0),(,k,"for x=0"):}`
Since, f(x) is continuous at x=0
`therefore LHL=f(0)`
Now, `LHL=underset(x to 0^(-))lim f(x)=underset(h to 0)lim f(0-h)`
`=underset(h to 0)lim (log(1+2(0-h)sin""(pi)/(180^(@))(0-h))`
`=underset(h to 0) (log (1+2(0-h)sin""(pi)/(180^(@))(0-h))/((0-h)^(2))`
`=underset(h to 0) (log (1-2h){-sin""(pih)/(180)})/(h^(2))`
`=underset(h to 0)lim (-2) (log (1-2h))/(-2h))=x^(-) underset(h to 0)lim (sin""(pih)/(180))/((pih)/(180))xx(pi)/(180)`
`=(-2) xx(-1) xx 1 xx (pi)/(180) " "[therefore underset(x to 0)lim log ""(1+x)/(x)=1 and underset(x to 0)lim (sin x)/(x)=1]`
`=(pi)/(90)`