# 8.1.3: Infinite Limits

- Page ID
- 14813

# Infinite Limits

Geeks-R-Us sells titanium mechanical pencils to computer algorithm designers. In an effort to attract more business, they decide to run a rather unusual promotion:

"SALE!! The more you buy, the more you save! Pencils are now \(\ \$ \frac{12 x}{x-3}\) per dozen!"

If the trillionaire, Spug Dense, comes in and says he wants to buy as many pencils as Geeks-R-Us can turn out, what will the cost of the pencils approach as the order gets bigger and bigger?

# Infinite Limits

Sometimes, a function may not be defined at a particular number, but as values are input closer and closer to the undefined number, a **limit** on the output may not exist. For example, for the function * f*(

*) = 1/x (shown in the figures below), as*

*x**values are taken closer and closer to 0 from the right, the function increases indefinitely. Also, as*

*x**values are taken closer and closer to 0 from the left, the function decreases indefinitely.*

*x*We describe these limiting behaviors by writing

\(\ \begin{array}{l}

\lim _{x \rightarrow 0^{+}} \frac{1}{x}=+\infty \\

\lim _{x \rightarrow 0^{-}} \frac{1}{x}=-\infty

\end{array}\)

Sometimes we want to know the behavior of \(\ f(x)\) as * x* increases or decreases without bound. In this case we are interested in the end behavior of the function, a concept you have likely explored before. For example, what is the value of \(\ f(x)=1 / x\) as \(\ x\) increases or decreases without bound? That is,

\(\ \begin{array}{l}

\lim _{x \rightarrow+\infty} \frac{1}{x}=? \\

\lim _{x \rightarrow-\infty} \frac{1}{x}=?

\end{array}\)

As you can see from the graphs (shown below), as \(\ x\) decreases without bound, the values of \(\ f(x) = 1/x\) are negative and get closer and closer to 0. On the other hand, as \(\ x\) increases without bound, the values of \(\ f(x) = 1/x\) are positive and still get closer and closer to 0.

That is,

\(\ \begin{array}{l}

\lim _{x \rightarrow+\infty} \frac{1}{x}=0 \\

\lim _{x \rightarrow-\infty} \frac{1}{x}=0

\end{array}\)

# Examples

Example 1

Earlier, you were asked a question about buying a lot of pencils.

**Solution**

As Spug buys more and more pencils, the cost of each dozen will drop quickly at first, and level out after a while, approaching $12 per dozen.

You can see the effect on the graph here:

Example 2

Evaluate the limit by making a graph: \(\ \lim _{x \rightarrow 3^{+}} \frac{x+6}{x-3}\)

**Solution**

By looking at the graph:

We can see that as x gets closer and closer to 3 from the positive side, the output increases right out the top of the image, on its way to ∞

Example 3

Evaluate the limit: \(\ \lim _{x \rightarrow \infty} \frac{11 x^{3}-14 x^{2}+8 x+16}{9 x-3}\).

**Solution**

To evaluate polynomial function limits, a little bit of intuition helps. Let's think this one through.

First, note that since we are looking at what happens as \(\ x \rightarrow \infty\) most of the interesting stuff will happen as \(\ x\) gets * really* big.

On the top part of the fraction, as * x* gets truly massive, the \(\ 11 x^{3}\) part will get bigger much faster than either of the other terms. In fact, it increases so much faster than the other terms completely cease to matter at all once

*gets really monstrous. That means that the important part of the top of the fraction is just the \(\ 11 x^{3}\).*

*x*On the bottom, a similar situation develops. As * x* gets really, really big, the

*matters less and less. So the bottom may as well be just \(\ 9x\).*

*-3*That gives us \(\ \frac{11 x^{3}}{9 x}\) which reduces to \(\ \frac{11 x^{2}}{9}\)

Now we can more easily see what happens at the "ends." As * x* gets bigger and bigger, the numerator continues to get bigger faster than the denominator, so the overall output also increases.

\(\ \therefore \lim _{x \rightarrow+\infty} \frac{11 x^{3}-14 x^{2}+8 x+16}{9 x-3} \text { is }+\infty\)

Example 4

Evaluate \(\ \lim _{x \rightarrow 0} \frac{x+2}{x+3}\)

**Solution**

This one is easier than it looks! As x-->0, leaving just the fraction: *2/3*

Example 5

Make a graph to evaluate the limit \(\ \lim _{x \rightarrow \infty} \frac{1}{\sqrt{x}}\) and \(\ \lim _{x \rightarrow 0^{+}} \frac{1}{\sqrt{x}}\).

**Solution**

By looking at the image, we see that as * x* gets huge, so does \(\ \sqrt{x}\) which means that 1 is being divided by an ever-larger number, and the result is getting smaller and smaller.

\(\ \lim _{x \rightarrow \infty} \frac{1}{\sqrt{x}}=0\)

On the same image, we can see that as \(\ x\) gets closer and closer to zero, so does \(\ \sqrt{x}\) which means that 1 is being divided by an ever smaller number, and the result gets bigger and bigger.

\(\ \lim _{x \rightarrow 0^{+}} \frac{1}{\sqrt{x}} \text { is }+\infty\)

Example 6

Graph and evaluate the limit: \(\ \lim _{x \rightarrow 2^{+}} \frac{1}{x-2}\).

**Solution**

By looking at the image, we can see that as * x* gets closer and closer to 2 from the positive direction, 1 gets divided by smaller and smaller numbers, so the result gets larger and larger.

\(\ \lim _{x \rightarrow 2^{+}} \frac{1}{x-2} \text { is }+\infty\)

# Review

Evaluate the limits:

- \(\ \lim _{x \rightarrow 3^{-}} \frac{1}{x-3}\)
- \(\ \lim _{x \rightarrow-4^{+}} \frac{1}{x+4}\)
- \(\ \lim _{x \rightarrow-\left(\frac{8}{3}\right)^{+}} \frac{1}{3 x+8}\)
- \(\ \lim _{x \rightarrow-5^{+}} \frac{\left(x^{2}+11 x+30\right)}{x+5}\)
- \(\ \lim _{x \rightarrow-\infty} \frac{\left(x^{2}+11 x+30\right)}{x+5}\)
- \(\ \lim _{x \rightarrow \infty} \frac{-11 x^{3}+20 x^{2}+15 x-17}{-9 x^{3}+5 x^{2}-x-17}\)
- \(\ \lim _{x \rightarrow \infty} 13\)
- \(\ \lim _{x \rightarrow \infty} \frac{-2 x+18}{17 x-3}\)
- \(\ \lim _{x \rightarrow \infty} 15\)
- \(\ \lim _{x \rightarrow \infty}-5 x^{2}+5 x+14\)
- \(\ \lim _{x \rightarrow \infty} 7 x+12\)
- \(\ \lim _{x \rightarrow \infty}-3 x+13\)
- \(\ \lim _{x \rightarrow \infty} \frac{13 x-8}{19 x^{3}-11 x^{2}+x+4}\)
- \(\ \lim _{x \rightarrow \infty}-17 x+14\)
- \(\ \lim _{x \rightarrow \infty}-7 x^{2}-2 x-13\)

# Vocabulary

Term | Definition |
---|---|

limit |
A limit is the value that the output of a function approaches as the input of the function approaches a given value. |