Question

Question #3c650

Answer 1

The limit is zero.

Explanation:

As `x rarr 0^-, 1/x rarr -oo.`
For such x, `e^(1/x) rarr 0` and the denominator of the expression goes to 1.
Therefore
As `x rarr 0^-, x/(1 + e^(1/x)) rarr 0.`

As `x rarr 0^-+ 1/x rarr oo.`
For such x, `e^(1/x) rarr oo` and the denominator of the expression goes to `oo`.
Therefore
As `x rarr 0^+, 1/(1 + e^(1/x)) rarr 0,` and
as `x rarr 0^+, x/(1 + e^(1/x)) rarr 0.`

Since the two one-sided limits exist and are equal, the overall limit is zero.