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 has non-removable discontinuity at `x=0`?

Options

A. `f(x)=1/(1+2^(cotx))`
B. `f(x)=xsin(pi)/x`
C. `f(x)=1/(ln|x|)`
D. `f(x)=sin((|sinx|)/x)`

Answer

A, D

Explanation

(A) `f(x)=1/(ln|x|) LHL=0= RHL`
(B)` f(x)=xsin(pi)/x LHL=0=RHL`
(C) `f(x)=1/(1+2^(cotx)) f(0)=` not define
`LHL=1`
`RHL=0impliesLHL!=RHL`
(D) `f(x)=sin((|sinx|)/x)LHL` (at `x=0`) `=sin(-1)=-sin1`
`RHL` (at` x=0`) `=sin1`
`LHL!=RHL`