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Math Help - [SOLVED] Why is limit of natural log equal to ln of limit?

  1. #1
    Senior Member x3bnm's Avatar
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    [SOLVED] Why is limit of natural log equal to ln of limit?

    Why is that

    \lim     _{x\to -\infty}  \frac{x+1}{x-1} = \ln(\lim _{x\to -\infty}  \frac{x+1}{x-1})

    Can anyone kindly explain it?
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by x3bnm View Post
    Why is that

    \lim _{x\to -\infty} \frac{x+1}{x-1} = \ln(\lim _{x\to -\infty} \frac{x+1}{x-1})

    Can anyone kindly explain it?
    It isn't. The first one is 1 the second is 0[/tex]
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  3. #3
    Senior Member x3bnm's Avatar
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    Sorry the expression was wrong in my last post. Edited one is :

    \lim  _{b\to -\infty} (\ln(|\frac{b-1}{b+1}|) ) = \ln(\lim _{b\to -\infty} \frac{b-1}{b+1})

    I found them Thomas' Calculus solution book as part of solving a integral(Chapter 8.8 exercise9). But why are they equal?
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  4. #4
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    Because ln(x) is continous for every x>0
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  5. #5
    Senior Member x3bnm's Avatar
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    Because ln(x) is continous for every x > 0
    Can you elaborate on this?
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  6. #6
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    The function is not discontinuous on any interval greater than 0, or does not have any asymptotic holes on intervals greater than 0, simply look at the graph as x is > 0 and it continues to reach infinity in terms of limits

    Finding the graph is simple, just use a calculator or manually find the "sequence of terms" for each x, like x = 1, x = 2, x = 3...and so on to make the graph and youll see why
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  7. #7
    Senior Member x3bnm's Avatar
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    Thanks hjortur and RockHard. I get it now.
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  8. #8
    Super Member redsoxfan325's Avatar
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    One of the definitions/implications of continuity is that if you have a sequence \{x_n\} such that \lim_{n\to\infty}x_n=x, then \lim_{n\to\infty}f(x_n)=f(x)
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