8.1.3: Infinite Limits

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Mar 27, 2022
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- 8.1.2: One-Sided Limits
- 8.1.4: Polynomial Function Limits

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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(x) = 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.

f-d_a1c872907db931401e43c359d41397d7f69e00b102ea5b86fd75d5f5+IMAGE_TINY+IMAGE_TINY.jpg

f-d_c3826dde494658e63429a0b807381d0fdd1eea097d6a266123417fec+IMAGE_TINY+IMAGE_TINY.jpg

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.

f-d_7309088a49a9161fc50afc8d722ecebf0c9766b35ecd49c5f8b9dc00+IMAGE_TINY+IMAGE_TINY.png

f-d_0084d4e7b6fb07a636d0387fd7cd595878852eda612a5bf27d372d25+IMAGE_TINY+IMAGE_TINY.png

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:

f-d_0e5c44b73a4afe1d566afa190c43d4b4ccb80181b06436a838607e1a+IMAGE_TINY+IMAGE_TINY.png

Example 2

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

Solution

By looking at the graph:

f-d_2ed9cc3b6c1496d1c454ee027600d2bcffc6289da4f8eb10b963f617+IMAGE_TINY+IMAGE_TINY.png

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

On the bottom, a similar situation develops. As x gets really, really big, the -3 matters less and less. So the bottom may as well be just $\ 9x$ .

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.

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

f-d_ecdaafc4cb447edc924f6b1316561ba52409c2e0a1d3429d419bdfea+IMAGE_TINY+IMAGE_TINY.png

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.

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

f-d_e07efe198e9ee2b280196f3f92ac48bf8853a5c4d99e633b85503ec5+IMAGE_TINY+IMAGE_TINY.png

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.

f-d_8cf47c13f17f1acb89275c566befe39ffb76766bcaacc6e46d196ece+IMAGE_TINY+IMAGE_TINY.png

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

Review

Evaluate the limits:

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

Review (Answers)

To see the Review answers, open this PDF file and look for section 8.3.

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.

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Infinite Limits

Infinite Limits

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