Introduction

Welcome to the article about connecting limits at infinity and horizontal asymptotes. 

Based on previous knowledge, you have probably worked with limits at specific numbers, in order to see how a function would behave near that point. In this topic, we won’t focus on a specific value/number, rather we are going to look at the long behavior of functions when the value of $x$ gets larger and larger (positive or negative).

This is encapsulated by limits at infinity, which allow us to see how functions would behave at their ends, when they approach a horizontal asymptote. Even though you can’t plug in infinity for a value of $x$, taking the limit as $x$ gets larger and larger is similar to how you would take limits for specific values.

Limits at Infinity

Like we discussed in the introduction, a limit as $x$ approaches infinity describes the value attained by the function as $x$ is taken to the very right ($x$ approaches $\infty$) or as $x$ is taken to the very left ($x$ approaches $-\infty$). 

Example 1:

Calculate $\underset{x\to \infty }{\lim }{e}^x$.

As $x$ gets really big, $e^x$ would also get big as exponential functions grow really fast. $e$ to the power of some number is a bigger number.

Thus, $\underset{x\to \infty }{\lim }{e}^x=\infty$. This describes the right end behavior of $e^x$. 

This is an elementary example, but it is a benchmark to more complex expressions such as solving the limit as $x$ approaches infinity of $e^x\ln x$. 

As a helping hand, it is useful to know that when dealing with polynomials, you should look at the highest degree of power and determine whether it’s even or odd, and whether the coefficient is positive or negative. This would help you quickly find out the limits of such polynomials by knowing their end behaviors.

[Editor's Note: $\mathbb{W}$ represents the mathematical set of whole numbers, that is $0,1,2,\mathrm{3...}$]

Even/Positive: Think of $x^{\left(2n\right)},n\in \mathbb{W}$ 

Even/Negative: Think of $-x^{\left(2n\right)},n\in \mathbb{W}$

Odd/Positive: Think of $x^{\left(2n+1\right)},n\in \mathbb{W}$

Odd/Negative: Think of $-x^{\left(2n+1\right)},n\in \mathbb{W}$

Example 2a: 

Calculate $\underset{x\to \infty }{\lim }\frac{\left(2x^5-x^2\right)}{x}$.

First, we see that we can go through with the division, to get $2x^4-x$. 

Then, we see that the biggest degree is $4$, and a positive coefficient (think  $x^{\left(2n\right)}n\in \mathbb{W}$), so the limit as $x$ approaches $\infty$ would be $\infty$.

Example 2b:

Calculate $\underset{x\to -\infty }{\lim }\frac{\left(2x^5-x^2\right)}{x}$

This is the same expression as in example 2a, but $x$ now approaches $-\infty$. Based on our analysis from before, we see that this follows a positive coefficient and even degree function, so the limit as $x$ approaches $-\infty$ would be $\infty$.

The example above hints at the more complex limits that we can see in our exploration of infinite limits, a concept known as indeterminate limits. 

Example 3:

Calculate $\underset{x\to \infty }{\lim }\frac{\left(x^{100}-93x^{53}+2\right)}{e^x}$.

If we take the limit as $x$ tends to infinity for the top and bottom functions, we see that both of them go towards infinity. This is one example of an indeterminate form where it is in the form $\frac{\infty }{\infty }$. Another such case is when the limit is in the form $\frac{0}{0}$. 

Because we don’t attain a single value, the limit will be determined by the one that has the greatest magnitude of growth. When comparing the exponential function $e^x$ to the function $x^{100}-93x^{53}+2$, we see that the exponential function grows at a greater rate compared to the polynomial (this is always the case when comparing between exponential and polynomial). Thus, we know that the bottom function in the original expression would shoot up to infinity faster than the top function, so the $\underset{x\to \infty }{\lim }\frac{\left(x^{100}-93x^{53}+2\right)}{e^x}$ would be $0$. 

If the denominator and numerator were swapped, then the exponential would be at the top, so the limit would go towards infinity, and if they grew at the same rate, then the limit would be a constant (not necessarily $1$ because of coefficients of the functions), which would determine the horizontal asymptote. 

In this case, the horizontal asymptote of the whole function in the limit expression is $0$.

Practice