If function
`f(x)=x-(|x|)/(x),x lt 0`
`=x+(|x|)/(x),x gt 0`
`=1,x=0`, then

Options

A. `underset(x to 0^(-))(lim) `f(x) dose not exist
B. `underset(x to 0^(+))(lim) `f(x) dose not exist
C. f(x) is continous at x = 0
D. `underset(x to 0^(-))(lim) f(x) ne underset(x to 0^(+))(lim) f(x)`

Answer

C

Explanation

Given function,
`f(x) = x - (|x|)/(x), x lt 0`
`= x + (|x|)/(x), x gt 0 = 1, x = 0`
Now, at x = 0
`LHL = underset(x to 0)(lim) f(x) = underset(x to 0^(-))(lim) (x - (|x|)/(x))`
`underset(x to 0)(lim) (x - ((-x))/(x)) = underset(x to 0)(lim) (x + 1) = 1`
`RHL = underset(x to 0^(+))(lim) f(x) = underset(x to 0)(lim) + (x + (|x|)/(x))`
`= underset(x to 0)(lim) (x + 1) = 1` and f(0) = 1
`:. underset(x to 0^(-))(lim) f(x) = underset(x to 0)(lim) f(x) = f(0) = 1`
`implies f(x)` is continuous at x = 0