1.15 Connecting Limits at Infinity and Horizontal Asymptotes

Limits at infinity describe a function's end behavior: what the output approaches as $x$ gets very large in the positive or negative direction. If $\lim_{x \to \infty} f(x) = L$ or $\lim_{x \to -\infty} f(x) = L$, then $y = L$ is a horizontal asymptote. For AP Calculus, connect the limit statement to the graph's far-left or far-right behavior.

Limits at infinity describe a function's end behavior: what the output approaches as $x$ gets very large in the positive or negative direction. If $\lim_{x \to \infty} f(x) = L$ or $\lim_{x \to -\infty} f(x) = L$, then $y = L$ is a horizontal asymptote.

This is different from an infinite limit near a vertical asymptote. Here, $x$ is moving toward infinity, and you are asking what $y$ approaches on the far left or far right of the graph.

🔍 Review: Limits to Infinity

What happens to a function as a variable gets very small or very large? We say that this variable is “approaching negative infinity or infinity.” By finding the limit as $x$ approaches infinity, we are trying to find what the y value is approaching as the x value continues to increase.

1. $\lim_{x\to\ \infty} f(x)$

2. $\lim_{x\to\ -\infty} f(x)$

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🧐 Finding Limits to Infinity

So, how do you find the limit as $x$ approaches infinity or negative infinity? You need to find the horizontal asymptote!

🚥 Horizontal Asymptotes

A horizontal asymptote is a y value that the graph cannot touch. 🛑 The function will continue to get closer to the horizontal asymptote but will never touch or cross it. Let’s look at an example:

On the graph, there is a horizontal asymptote at $y = 5$. The function cannot cross the graph at that point. Therefore, $\lim_{x \to \infty} f(x) = 5$.

🔍 Finding Horizontal Asymptotes

There are a few rules to follow when finding the horizontal asymptote (and in turn, the limit at infinity) of a function. In order to apply the rules, think of a function as $\frac{p(x)}{q(x)}$.

🥇 If the degree of p(x) < the degree of q(x), then the HA (Horizontal Asymptote) = 0.

$\lim_{x \to \infty} \frac{4x+2}{3x^2+1} = 0$

🥈 If the degree of p(x) > the degree of q(x), then there is no horizontal asymptote.

$\lim_{x \to \infty} \frac{4x^2+2}{3x+1} = \infty$

🥉 If the degree of p(x) = the degree of q(x), then the HA is the ratio of the coefficients.

$\lim_{x \to \infty} \frac{3x^2+2}{4x^2+1} = \frac{3}{4}$

Finding the HA can be used to find the limit when the function is indeterminate, meaning that you have $\frac {\infty}{\infty}$. There are also some other ways to find limits approaching infinity.

Exceptions to These Rules

• The limit of an oscillating function at infinity does not exist.
  • Ex: $\lim_{x \to \infty}\sin x = DNE$
• Any function that is part of the squeeze theorem will equal 0 as it approaches infinity.
  • Ex: $\lim_{x \to \infty}\frac {\sin x}{x} = 0$

🌱 The Growth Rates of Functions

It is important to note that not all functions grow, 🪴 or increase, at the same rate. The order of the growth rates of common functions is as follows, from slowest to fastest:

• log < root < polynomial < exponential.

Knowing the growth rates of each function can be helpful when determining its limit at infinity.

✏️ Infinite Limits & Horizontal Asymptotes Practice

Infinite Limits: Example 1

Evaluate the following limit:

$\lim_{x \to \infty}\frac {3x-1}{e^x}$

On the bottom of the fraction there is an exponential function, $e^x$. Exponential functions have very high growth rates, so the denominator of the function will get increasingly larger while the numerator stays relatively small. Therefore it is like having a $\frac {\text small \,number}{large \, number}$.

Since the number on the denominator is so much larger than the number in the numerator, when divided, the quotient = 0. Therefore $\lim_{x \to \infty}\frac {3x-1}{e^x} = 0$.

Infinite Limits: Example 2

Evaluate the following limit:

In this problem, since the exponential function is in the numerator, we have a $\frac {\text large\,number}{small \, number}$. Therefore, $\lim_{x \to \infty}\frac {2^x}{x-5} = \infty$. When you have a function that grows to be so large, so fast, all the other numbers are negligible comparatively!