How to evaluate continuity of this function?

Question

f(x) = {((e^1/x )/(1+e^1/x)) if x ≠ 0

{1 if x= 0

Now I can either directly put x=0 in left and right hand limits after evaluating

f(0) = 1.In which case 1/x approaches infinity and function becomes discontinuous at x= 0

Or I can put x = 0 - h for left and limit and 0 + h for right hand limit in which case it's still discontinuous if you solve it.

So which is correct the first method or the second one or are both

correct or wrong?

Answer 1

The first method is incorrect for 2 reasons:

(a) If 1/x approaches to infinity, it does not mean yet that f(x) becomes discontinuous

(b) Actually, 1/x does not approach to infinity. It approaches to infinity on the right and to negative infinity on the left.

But the second method is correct. You should find 2 limits: when x approaches to 0+ and when x approaches to 0-. (You claimed that you found these limits.)

If the both limits were equal to 1, f(x) will be continuous function. Otherwise, it is not continuous at x=0.