# 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:

 $\ds \lim_{x \mathop \to 0^+} \map f x$ $=$ $\ds 0$ $\ds \lim_{x \mathop \to 0^-} \map f x$ $=$ $\ds 1$

## Proof

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.

$\Box$

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 $1$.

$\blacksquare$