Exponential Tends to Zero and Infinity

Theorem
Let $x \in \R$ be a real number.

Let $\exp x$ be the exponential of $x$.

Then:
 * $\exp x \to +\infty$ as $x \to +\infty$
 * $\exp x \to 0$ as $x \to -\infty$

Thus the exponential function has domain $\R$ and image $\left({0 \,.\,.\, +\infty}\right)$.

The exponential function approaches 0 as x approaches negative infinity
Let $\epsilon \in \R_{>0}$.

Let $c$ be $\operatorname{ln} \epsilon - 1$,

Let $x$ be a real number, where $x < c$,

By Exponential is Strictly Increasing and Convex:
 * $\displaystyle \exp x < \exp c$.

By the definition of $c$:
 * $\displaystyle \exp c = \exp \ln \epsilon - 1 = \frac {\exp \ln \epsilon} e = \frac \epsilon e$.

Therefore:
 * $\displaystyle \exp x < \frac \epsilon e < \epsilon$.

Hence:
 * $\displaystyle \therefore \forall \epsilon \in \R_{>0} : \exists c : \forall x > c : \exp x < \epsilon$.

From the definition of limit involving infinity, the result follows.

The exponential function has domain $\R$ and image $\left({0 \,.\,.\, +\infty}\right)$
We have that the Exponential is Strictly Increasing and Convex.

From above, $\displaystyle \lim_{x \to \infty} \exp x = \infty$

From above, $\displaystyle \lim_{x \to -\infty} \exp x = 0$

Hence the result.