Sometimes when he have limits, they don't really approach a point, they just keep growing to infinity (or negative infinity)

Back in Pre-Calc, we called these asymptotes, but with calculus we can define these using limits to infinity ($\infty$)

Limits With Infinities (assume $c$ is a number):

$$\begin{align} \lim_{x \to \infty} f(x) = c \\ \lim_{x \to -\infty} f(x) = c \\ \lim_{x \to c} f(x) = \infty \\ \lim_{x \to c} f(x) = -\infty \end{align}$$

The first two limits are approaching an infinity

The other two limits are resulting in an infinity

Let's look at some example graphs

$$\begin{align} \lim_{x \to \infty} f(x) = 0 \\ \lim_{x \to -\infty} f(x) = 0 \\ \end{align}$$

$$\begin{align} \lim_{x \to 2} f(x) = \infty \\ \end{align}$$

$$\begin{align} \lim_{x \to -\infty} f(x) = 1 \\ \lim_{x \to \infty} f(x) = \infty \\ \end{align}$$

How Do We Solve Limits At Infinity?

For limits that result in infinity, they come about with asymptotes or unbounded growth

By Unbounded growth, think of a function like $f(x) = 3x$, as x gets infintly large, so does $f(x)$

There's nothing stopping the size of the function, so it just keeps growing forever

The main way to solve limits approaching infinity is to think about growth

Which function grows faster (when $x$ is very large), $x^2$ or $x^{90}$

It's the $x^{90}$ of course, because it multiplies it self more

The main idea with solving limits approaching infinties, is that the biggest, or fastest growing function dominates

For example:

$$\begin{align} \lim_{x \to \infty} 2^xx^2 \end{align}$$

When $x$ is large, the $2^x$ grows much faster, so it overtakes the result of $x^2$

Now suppose the two function $4x^3 + 3x$ and $5x^3$

Which of these two functions grows faster?

They both grow at the rate $x^3$ (the $3x$ doesn't matter because the $x^3$ grows much faster at large $x$ values)

Since they both grow at the same rate, you need to then compare the coefficients because those are differnet

The $5x^3$ grows faster because $5 > 4$

Basically, if the growth is same, then you need to focus on their coefficients

Infinity Properties (In Limits)

If you get to these in a limit approaching infinity, you can apply these

$$\begin{align} \frac{1}{\infty} &= 0 \\ \frac{1}{0^+} &= \infty \\ \frac{1}{0^-} &= -\infty \\ \end{align}$$

This is because when you divide by bigger numbers, that means you are splitting $1$ into portions

If you have more portions, the size of each portion gets smaller

At infinite portions, the size of each portion goes to 0

$$\begin{align} \infty + \infty &= \infty \\ \infty \cdot \infty &= \infty \\ c \cdot \infty &= \infty &&\text{$c$ is a number}\\ \infty - \infty &= \text{Indeterminate} \\ \end{align}$$

The first two are because something very large times/plus another thing very large is also very large

Negative and positive infinity still follow the same properties as positive and negative numbers

$$\begin{align} -\infty &= -1 \cdot \infty\\ \infty \cdot \infty &= \infty \\ \infty \cdot -\infty &= -\infty \\ -\infty \cdot \infty &= -\infty \\ -\infty \cdot -\infty &= \infty \\ \end{align}$$

Rational Functions

For limits approaching infinity with rational functions

The growth of the numerator and denominator needs to be considered

If the numerator grows faster, then it results in an infinity

If the denominator grows faster, then it results at 0

If they both grow at the same rate, then you need to divide the coefficients for the fastest growing terms

This is because since they grow at the same rate, they must be proportional ($f(x) \propto g(x)$)

For polynomials, it is easy to find the fastest growing function, whichever has the higher degree (largest exponent)

For example:

$$\begin{align} \lim_{x \to \infty} \frac{x^3 - 4x + 2}{x^2 - 9} &= \infty \\ \lim_{x \to \infty} \frac{x^2 - \sqrt{x}}{x^3 - 9x + 4} &= 0 \\ \lim_{x \to \infty} \frac{3x^2 + \sin(x)}{4x^2 + \cos(x)} &= \frac{3}{4} \\ \end{align}$$

For the last limit, the $\sin(x)$ and $\cos(x)$ don't matter because they are bounded on $[-1,1]$, which means they can't grow infintly

If a function is bounded it also means it has a fixed range

AVOID INDETERMINATE FORMS!

Read up on Indeterminate Forms and make sure to avoid them

The forms concering infinity are:

$$\begin{align} \frac{\infty}{\infty},\infty - \infty, 0 \cdot \infty, 1^{\infty}, \infty^0 \end{align}$$

Practice Problems