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Math Help - Quick and likely easy limit theory question

  1. #1
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    Quick and likely easy limit theory question

    Hi all,

    I just have a quick general question regarding limits. Say you have the problem:

    \[\mathop {\lim }\limits_{x \to \infty } \sqrt x {e^{ - 5x}}\]

    My calculator says it's 0, and others have declared similar problems as 0. Looking at it, though, wouldn't you be evaluating \[\infty  \times 0\], which would be indeterminate?

    Why is it 0?
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  2. #2
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    Quote Originally Posted by Malaclypse View Post
    Hi all,

    I just have a quick general question regarding limits. Say you have the problem:

    \[\mathop {\lim }\limits_{x \to \infty } \sqrt x {e^{ - 5x}}\]

    My calculator says it's 0, and others have declared similar problems as 0. Looking at it, though, wouldn't you be evaluating \[\infty  \times 0\], which would be indeterminate?

    Why is it 0?
    \displaystyle \lim_{x \to \infty} \frac{\sqrt{x}}{e^{5x}}

    as x \to \infty , e^{5x} >>>>>>> \sqrt{x}

    the limit can also be shown to be 0 using L'Hopital
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  3. #3
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    Here is another to look at it:
    \displaystyle \frac{\sqrt{x}}{e^{5x}}<\frac{e^x}{e^{5x}}=\frac{1  }{e^{4x}}
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  4. #4
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    I'm fairly new to a lot of this, so thank you very much...and the fact that it's self-study makes it a bit more difficult. The more I work with this stuff the more I realize that it's all about manipulating the equations when you hit a roadblock. I'm just the type of person that can't accept that it's "0 because the calculator says it's 0" or because somebody else says so, though...I need to see why and how, and then put it to practice.

    I guess the comforting thing to me in it all is that I've had 0 problem with calculus...it's just the algebra I keep forgetting!

    For me it's the simple things. The reaaaaaaaaaally simple things that I forget when I'm doing calculus. Things like "Hey, because that's a negative exponent I can just shift that to the denominator".... L'Hopital works perfectly then.

    Anyway, though...thanks for taking the time you two. Easy question, I know, but I always want to be sure I understand the fundamentals and make sure I take nothing for granted.
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