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Why does Lim as x approaches infinity of x/(x-9) = 1?

  1. Feb 8, 2012 #1
    1. The problem statement, all variables and given/known data

    Limit as x approaches infinity of [itex]x/x-9[/itex]

    2. Relevant equations

    None

    3. The attempt at a solution

    I know the indeterminate form infinity/infinity happens. I don't know how to fix it, but I'm assuming it's quite simple...
     
  2. jcsd
  3. Feb 8, 2012 #2
    When x is very very large, say 6 billion. The numbers 6 billion and 6 billion minus 9 are essentially the same. As x gets larger and larger, this discrepancy diminishes. The value the ratio approaches is 1.

    If you wanted to be a little more rigorous, you can divide the numerator and denominator by 1/x then take the limit and see what happens. It should pop right out.
     
  4. Feb 9, 2012 #3
    A nice trick for these types of problems would be to divide everything by x;

    [tex] \lim_{x\rightarrow\infty} \frac{x/x}{(x/x)-(9/x)} [/tex]

    Then simplify and you should be able to figure it out from there
     
  5. Feb 9, 2012 #4
    Using L'Hopitals rule for evaluating indeterminate forms, just differentiate both the numerator and the denominator separately to produce 1/1. Xyius explains well intuitively why the result is what it is.
     
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