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

Proof
We have that the exponential function is the inverse of the natural logarithm function:

The domain of $\ln x$ is domain $\left({0 .. \infty}\right)$ and its image is $\R$, hence the result.

The result follows from the definition of an inverse mapping.