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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(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:

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

    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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