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Infinite Limits

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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{12x}{x - 3} 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

Embedded Video:

- 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

\lim_{x \rightarrow 0^+} \frac{1} {x} = + \infty
\lim_{x \rightarrow 0^-} \frac{1} {x} = - \infty

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,

\lim_{x \to +\infty} \frac{1} {x} = ?
\lim_{x \to -\infty} \frac{1} {x} = ?

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,

\lim_{x \to +\infty} \frac{1} {x} = 0
\lim_{x \to -\infty} \frac{1} {x} = 0

Example A

Evaluate the limit by making a graph:

\lim_{x\to 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 \infty

Example B

Evaluate the limit \lim_{x\to \infty} \frac{11x^3 - 14x^2 +8x +16} {9x - 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 \to \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 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 \frac{11x^3}{9x} which reduces to \frac{11x^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\to +\infty} \frac{11x^3 - 14x^2 +8x +16} {9x - 3} = +\infty

Example C

Evaluate \lim_{x\rightarrow 0} \frac{x + 2} {x + 3}

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:

Vocabulary

A function with an infinite limit continues to output greater and greater +/- values.

Guided Practice

Questions

1) Make a graph to evaluate the limit \lim_{x\rightarrow \infty} \frac {1} {\sqrt{x}}

2) From problem 1, evaluate \lim_{x\rightarrow 0^{+}} \frac {1} {\sqrt{x}}

3) Graph and evaluate the limit:

\lim_{x\to 2^{+}} \frac {1} {x - 2}

Solutions

1) 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.

The limit is 0

2) 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.

The limit is +\infty

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.

Practice

Evaluate the limits, you may graph if you wish:

  1. \lim_{x\to 3^{-}} \frac {1} {x - 3}
  2. \lim_{x\to -4^{+}} \frac {1} {x + 4}
  3. \lim_{x\to -\left(\frac{8}{3}\right)^{+}} \frac {1} {3x + 8}
  4. \lim_{x\to -5^{+}} \frac{\left(x^2+11x+30\right)}{x+5}
  5. \lim_{x\to -\infty} \frac{\left(x^2+11x+30\right)}{x+5}

Evaluate the limits

  1. \lim{x \to \infty} \frac{-11x^3 + 20x^2 + 15x - 17}{-9x^3 + 5x^2 - x - 17}=
  2. \lim{x \to \infty} 13 =
  3. \lim{x \to \infty} \frac{-2x + 18}{17x - 3}=
  4. \lim{x \to \infty} 15=
  5. \lim{x \to \infty} -5x^2 + 5x + 14=
  6. \lim{x \to \infty}7x + 12=
  7. \lim{x \to \infty} -3x + 13=
  8. \lim{x \to \infty} \frac{13x - 8}{19x^3 - 11x^2 + x + 4}=
  9. \lim{x \to \infty} -17x + 14=
  10. \lim{x \to \infty}-7x^2 - 2x - 13 =

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