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 \begin{align*}\$ \frac{12x}{x - 3}\end{align*}

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.

We describe these limiting behaviors by writing

\begin{align*}\lim_{x \rightarrow 0^+} \frac{1} {x} = + \infty\end{align*}

\begin{align*}\lim_{x \rightarrow 0^-} \frac{1} {x} = - \infty\end{align*}

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{align*}\lim_{x \to +\infty} \frac{1} {x} = ?\end{align*}

\begin{align*}\lim_{x \to -\infty} \frac{1} {x} = ?\end{align*}

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{align*}\lim_{x \to +\infty} \frac{1} {x} = 0\end{align*}

\begin{align*}\lim_{x \to -\infty} \frac{1} {x} = 0\end{align*}

### Examples

#### Example 1

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

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: \begin{align*}\lim_{x\to 3^+} \frac{x + 6}{x - 3}\end{align*}

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 \begin{align*}\infty\end{align*}

#### Example 3

Evaluate the limit: \begin{align*}\lim_{x\to \infty} \frac{11x^3 - 14x^2 +8x +16} {9x - 3}\end{align*}

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 \begin{align*}x \to \infty\end{align*}*x* gets *really* big.

On the top part of the fraction, as *x* gets truly massive, the *11x** ^{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

*11x*

*.*

^{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 \begin{align*}\frac{11x^3}{9x}\end{align*}

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.

\begin{align*}\therefore \lim_{x\to +\infty} \frac{11x^3 - 14x^2 +8x +16} {9x - 3}\text{ is }+\infty\end{align*}

#### Example 4

Evaluate \begin{align*}\lim_{x\rightarrow 0} \frac{x + 2} {x + 3} \end{align*}

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

#### Example 5

Make a graph to evaluate the limit \begin{align*}\lim_{x\rightarrow \infty} \frac {1} {\sqrt{x}} \end{align*}

By looking at the image, we see that as *x* gets huge, so does \begin{align*}\sqrt{x}\end{align*}

\begin{align*}\lim_{x\rightarrow \infty} \frac {1} {\sqrt{x}} \end{align*}

On the same image, we can see that as *x* gets closer and closer to zero, so does \begin{align*}\sqrt{x}\end{align*}

\begin{align*}\lim_{x\rightarrow 0^{+}} \frac {1} {\sqrt{x}} \end{align*}

#### Example 6

Graph and evaluate the limit: \begin{align*}\lim_{x\to 2^{+}} \frac {1} {x - 2}\end{align*}

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.

\begin{align*}\lim_{x\to 2^{+}} \frac {1} {x - 2}\text{ is }+\infty\end{align*}

### Review

Evaluate the limits:

- \begin{align*}\lim_{x\to 3^{-}} \frac {1} {x - 3}\end{align*}
limx→3−1x−3 - \begin{align*}\lim_{x\to -4^{+}} \frac {1} {x + 4}\end{align*}
limx→−4+1x+4 - \begin{align*}\lim_{x\to -\left(\frac{8}{3}\right)^{+}} \frac {1} {3x + 8}\end{align*}
limx→−(83)+13x+8 - \begin{align*}\lim_{x\to -5^{+}} \frac{\left(x^2+11x+30\right)}{x+5}\end{align*}
limx→−5+(x2+11x+30)x+5 - \begin{align*}\lim_{x\to -\infty} \frac{\left(x^2+11x+30\right)}{x+5}\end{align*}
limx→−∞(x2+11x+30)x+5 - \begin{align*}\lim_{x \to \infty}\end{align*}
limx→∞ \begin{align*}\frac{-11x^3 + 20x^2 + 15x - 17}{-9x^3 + 5x^2 - x - 17}\end{align*}−11x3+20x2+15x−17−9x3+5x2−x−17 - \begin{align*}\lim_{x \to \infty} 13 \end{align*}
limx→∞13 - \begin{align*}\lim_{x \to \infty} \frac{-2x + 18}{17x - 3}\end{align*}
limx→∞−2x+1817x−3 - \begin{align*}\lim_{x \to \infty} 15\end{align*}
limx→∞15 - \begin{align*}\lim_{x \to \infty} -5x^2 + 5x + 14\end{align*}
limx→∞−5x2+5x+14 - \begin{align*}\lim_{x \to \infty}7x + 12\end{align*}
limx→∞7x+12 - \begin{align*}\lim_{x \to \infty} -3x + 13\end{align*}
limx→∞−3x+13 - \begin{align*}\lim_{x \to \infty} \frac{13x - 8}{19x^3 - 11x^2 + x + 4}\end{align*}
limx→∞13x−819x3−11x2+x+4 - \begin{align*}\lim_{x \to \infty} -17x + 14\end{align*}
- \begin{align*}\lim_{x \to \infty}-7x^2 - 2x - 13\end{align*}

### Review (Answers)

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