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Infinite Limits

Limits where the input or output approaches positive or negative infinity.

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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 \begin{align*}\$ \frac{12x}{x - 3}\end{align*} per dozen!"

If the zillionaire 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?

Watch This

James Sousa: Limits at Infinity

Guidance

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,

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

Example A

Evaluate the limit by making a graph:

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

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

Example B

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

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 @$\begin{align*}x \to \infty\end{align*}@$ 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 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 @$\begin{align*}\frac{11x^3}{9x}\end{align*}@$ which reduces to @$\begin{align*}\frac{11x^2}{9}\end{align*}@$

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.

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

Example C

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

Solution:

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

Concept question wrap-up:

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:

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Guided Practice

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

2) From problem 1, evaluate @$\begin{align*}\lim_{x\rightarrow 0^{+}} \frac {1} {\sqrt{x}} \end{align*}@$

3) Graph and evaluate the limit:

@$\begin{align*}\lim_{x\to 2^{+}} \frac {1} {x - 2}\end{align*}@$

Answers

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

The limit is 0

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

The limit is @$\begin{align*}+\infty\end{align*}@$

3) 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.

Explore More

Evaluate the limits:

  1. @$\begin{align*}\lim_{x\to 3^{-}} \frac {1} {x - 3}\end{align*}@$
  2. @$\begin{align*}\lim_{x\to -4^{+}} \frac {1} {x + 4}\end{align*}@$
  3. @$\begin{align*}\lim_{x\to -\left(\frac{8}{3}\right)^{+}} \frac {1} {3x + 8}\end{align*}@$
  4. @$\begin{align*}\lim_{x\to -5^{+}} \frac{\left(x^2+11x+30\right)}{x+5}\end{align*}@$
  5. @$\begin{align*}\lim_{x\to -\infty} \frac{\left(x^2+11x+30\right)}{x+5}\end{align*}@$
  6. @$\begin{align*}\lim_{x \to \infty}\end{align*}@$ @$\begin{align*}\frac{-11x^3 + 20x^2 + 15x - 17}{-9x^3 + 5x^2 - x - 17}\end{align*}@$
  7. @$\begin{align*}\lim_{x \to \infty} 13 \end{align*}@$
  8. @$\begin{align*}\lim_{x \to \infty} \frac{-2x + 18}{17x - 3}\end{align*}@$
  9. @$\begin{align*}\lim_{x \to \infty} 15\end{align*}@$
  10. @$\begin{align*}\lim_{x \to \infty} -5x^2 + 5x + 14\end{align*}@$
  11. @$\begin{align*}\lim_{x \to \infty}7x + 12\end{align*}@$
  12. @$\begin{align*}\lim_{x \to \infty} -3x + 13\end{align*}@$
  13. @$\begin{align*}\lim_{x \to \infty} \frac{13x - 8}{19x^3 - 11x^2 + x + 4}\end{align*}@$
  14. @$\begin{align*}\lim_{x \to \infty} -17x + 14\end{align*}@$
  15. @$\begin{align*}\lim_{x \to \infty}-7x^2 - 2x - 13\end{align*}@$

Vocabulary

limit

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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