One-Sided Limit of Real Function/Examples/Reciprocal of 1 + e to the Reciprocal of x

Examples of One-Sided Limits of Real Functions
Let $f: \R \to \R$ be the real function defined as:
 * $\map f x = \dfrac 1 {1 + e^{1 / x} }$

Then:

Proof

 * Reciprocal-1+e-to-1-over-x.png

We have:
 * $\dfrac 1 x \to +\infty$ as $x \to 0^+$

and so:
 * $1 + e^{1/x} \to +\infty$ as $x \to 0^+$

Thus:
 * $\dfrac 1 {1 + e^{1 / x} } \to 0$ as $x \to 0^+$

and remains positive.

Then we have:
 * $\dfrac 1 x \to -\infty$ as $x \to 0^-$

and so:
 * $-e^{1/x} \to 0$ as $x \to 0^-$

Thus:
 * $\dfrac 1 {1 + e^{1 / x} } \to 1$ as $x \to 0^-$

and remains less than $1$.