which of the following function(s) not ...

which of the following function(s) not defined at `x=0` has/have removable discontinuity at `x = 0`.

`f(x) = (1)/(1+2^(cot x))`

`f(x) = "cos"(((|sin x|))/(x))`

`f(x) = "x sin"(pi)/(x)`

`f(x) = (1)/("In"|x|)`

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The correct Answer is:

B, C, D

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If both Lim_(xrarrc^(-))f(x) and Lim_(xrarrc^(+))f(x) exist finitely and are equal, then the function f is said to have removable discontinuity at x=c . If both the limits i.e. Lim_(xrarrc^(-))f(x) and Lim_(xrarrc^(+))f(x) exist finitely and are not equal, then the function f is said to have non-removable discontinuity at x=c . Which of the following function not defined at x=0 has removable discontinuity at the origin?

`f(x)=1/(1+2^(1/x))`

`f(x)="tan"^(-1) 1/x`

`f(x)=(e^(1/x)-1)/(e^(1/x)+1)`

`f(x)=(|sinx|)/(|x|)`

`f(x) =(x^(2)-2x-8)/(x+2)at x=-2`

`f(x) =(x-7)/(|x-7|)at x=7`

`f(x) =(x^(3)+64)/(|x+4|)at x=-4`

`f(x) =(3-sqrt(x))/(9-x)at x=9`