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 $\left({0 \,.\,.\, +\infty}\right)$.

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

When $c = \operatorname{ln} \epsilon - 1$,

For all $x < c$,

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.