If the function `f(x)={(cosx)^(1/x),x!=0k ,x=0` is continuous at `x=0` , then the value of `k` is `0` (b) `1` (c) `-1` (d) `e`

Explanation

For f(x) to be continuous at x=0, we must have

`lim​_(x→0)f(x)=f(0)`

`⇒lim​_(x→0)(cosx)^(1/x)=k`

`⇒lim​_(x→0)(1+(cosx−1))^(1/x)=k`

`⇒e^(lim​_(x→0)​(cosx−1)/x)​=k`

`⇒e^(−lim​_(x→0)​(2sin^2(x/2))/x​​=k`

`⇒e^(−lim​_(x→0)​(​sin^2(x/2​)​)/(x/2)=k`

`⇒e^0=k`

...