If the function f(x) defined by
`f(x)={(x"sin"(1)/(x)",", "for "x ne 0),(k",", "for " x=0):}`
is continuous at x = 0, then k is equal to

Options

A. 0
B. 1
C. `-1`
D. `(1)/(2)`

Answer

A

Explanation

Given, `f(x)={{:(,x sin""(1)/(x), "for x" ne 0),(,k, "for x=0"):}`
Since, f(x) is continuous at x=0.
`therefore LHL=RH=f(0)`
Now, `LHL=underset(x to 0)lim f(x)=underset(h to 0)lim f(0-h)`
`underset(h to 0)lim (0-h) sin (1)/(0-h))=underset(h to 0)lim -h sin (-(1)/(h))`
`underset(h to 0)lim h sin (1)/(g)=0 xx` finite value `[therefore underset(h to 0)lim sin""(1)/(h)="finite value"]`
`and f(0)=k`
`therefore "From Eq. (i) LHL=f(0)"`
`Rightarrow 0=k`
brgt `Rightarrow k=0`