# Limits at Infinity and Horizontal Asymptotes

## Key Questions

• Example 1

${\lim}_{x \to \infty} \frac{x - 5 {x}^{3}}{2 {x}^{3} - x + 7}$

by dividing the numerator and the denominator by ${x}^{3}$,

$= {\lim}_{x \to \infty} \frac{\frac{1}{x} ^ 2 - 5}{2 - \frac{1}{x} ^ 2 + \frac{7}{x} ^ 3} = \frac{0 - 5}{2 - 0 + 0} = - \frac{5}{2}$

Example 2

${\lim}_{x \to - \infty} x {e}^{x}$

since $- \infty \cdot 0$ is an indeterminate form, by rewriting,

$= {\lim}_{x \to - \infty} \frac{x}{e} ^ \left\{- x\right\}$

by l'Hopital's Rule,

$= {\lim}_{x \to - \infty} \frac{1}{- {e}^{- x}} = \frac{1}{- \infty} = 0$

I hope that this was helpful.

Another perspective...

#### Explanation:

$\textcolor{w h i t e}{}$
As a Real function

Treating ${e}^{x}$ as a function of Real values of $x$, it has the following properties:

• The domain of ${e}^{x}$ is the whole of $\mathbb{R}$.

• The range of ${e}^{x}$ is $\left(0 , \infty\right)$.

• ${e}^{x}$ is continuous on the whole of $\mathbb{R}$ and infinitely differentiable, with $\frac{d}{\mathrm{dx}} {e}^{x} = {e}^{x}$.

• ${e}^{x}$ is one to one, so has a well defined inverse function ($\ln x$) from $\left(0 , \infty\right)$ onto $\mathbb{R}$.

• ${\lim}_{x \to + \infty} {e}^{x} = + \infty$

• ${\lim}_{x \to - \infty} {e}^{x} = 0$

At first sight this answers the question, but what about Complex values of $x$?

$\textcolor{w h i t e}{}$
As a Complex function

Treated as a function of Complex values of $x$, ${e}^{x}$ has the properties:

• The domain of ${e}^{x}$ is the whole of $\mathbb{C}$.

• The range of ${e}^{x}$ is $\mathbb{C} \text{\} \left\{0\right\}$.

• ${e}^{x}$ is continuous on the whole of $\mathbb{C}$ and infinitely differentiable, with $\frac{d}{\mathrm{dx}} {e}^{x} = {e}^{x}$.

• ${e}^{x}$ is many to one, so has no inverse function. The definition of $\ln x$ can be extended to a function from $\mathbb{C} \text{\} \left\{0\right\}$ into $\mathbb{C}$, typically onto $\left\{x + i y : x \in \mathbb{R} , y \in \left(- \pi , \pi\right]\right\}$.

What do we mean by the limit of ${e}^{x}$ as $x \to \text{infinity}$ in this context?

From the origin, we can head off towards "infinity" in all sorts of ways.

For example, if we just set off along the imaginary axis, the value of ${e}^{x}$ just goes round and around the unit circle.

If we choose any complex number $c = r \left(\cos \theta + i \sin \theta\right)$, then following the line $\ln r + i t$ for $t \in \mathbb{R}$ as $t \to + \infty$, the value of ${e}^{\ln r + i t}$ will take the value $c$ infinitely many times.

We can project the Complex plane onto a sphere called the Riemann sphere ${\mathbb{C}}_{\infty}$, with an additional point called $\infty$. This allows us to picture the "neighbourhood of $\infty$" and think about the behaviour of the function ${e}^{x}$ there.

From our preceding observations, ${e}^{x}$ takes every non-zero complex value infinitely many times in any arbitrarily small neighbourhood of $\infty$. That is called an essential singularity at infinity.