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Math Help - limit as x -> infty

  1. #1
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    limit as x -> infty

    \lim_{x \rightarrow \infty}(x-lnx)

    I made it into indeterminate form so that I can use l'Hospital's rule:
    \lim_{x \rightarrow \infty}(lne^x-lnx) = \lim_{x \rightarrow \infty}ln\left(\frac{e^x}{x}\right)

    I take the derivative and get:

    \frac{x-1}{x}

    Taking the derivative again I get that the function approaches 1, but based off of the actual graph the function approaches infinity. What am I doing wrong?
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  2. #2
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    Re: limit as x -> infty

    \lim_{x\to\infty}\ln\left(\frac{e^x}{x} \right)=\ln\left(\lim_{x\to\infty}\frac{e^x}{x} \right)

    Now apply L'H˘pital's rule to the limit inside the log function.
    Thanks from amthomasjr
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  3. #3
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    Re: limit as x -> infty

    Hello, amthomasjr!

    \lim_{x \rightarrow \infty}(x-\ln x)

    I made it into indeterminate form so that I can use l'Hospital's rule:

    \lim_{x\to\infty}(\ln e^x-\ln x) \:=\: \lim_{x\to\infty}\ln\left(\frac{e^x}{x}\right)

    I take the derivative and get: . \frac{x-1}{x} . How?

    Exactly what did you differentiate?

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  4. #4
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    Re: limit as x -> infty

    Quote Originally Posted by Soroban View Post
    Hello, amthomasjr!


    Exactly what did you differentiate?

    ln\left(\frac{e^x}{x}\right)
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  5. #5
    MHF Contributor MarkFL's Avatar
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    Re: limit as x -> infty

    In order to apply L'H˘pital's rule you need the limit to take the form:

    \lim_{x\to c}\frac{f(x)}{g(x)}
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  6. #6
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    Re: limit as x -> infty

    Quote Originally Posted by MarkFL2 View Post
    \lim_{x\to\infty}\ln\left(\frac{e^x}{x} \right)=\ln\left(\lim_{x\to\infty}\frac{e^x}{x} \right)

    Now apply L'H˘pital's rule to the limit inside the log function.
    That makes sense, so

    ln\left(e^x\right) = x and

    \lim_{x\to\infty}x = \infty
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  7. #7
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    Re: limit as x -> infty

    Quote Originally Posted by MarkFL2 View Post
    In order to apply L'H˘pital's rule you need the limit to take the form:

    \lim_{x\to c}\frac{f(x)}{g(x)}
    Which is why you only apply it to the inside, correct?
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  8. #8
    MHF Contributor MarkFL's Avatar
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    Re: limit as x -> infty

    I would write:

    \ln\left(\lim_{x\to\infty}\frac{e^x}{x} \right)=\ln\left(\lim_{x\to\infty}e^x \right)=\ln(\infty)=\infty
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  9. #9
    MHF Contributor MarkFL's Avatar
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    Re: limit as x -> infty

    Quote Originally Posted by amthomasjr View Post
    Which is why you only apply it to the inside, correct?
    Right, on the inside of the log function, you now have the correct form to use the rule bought by L'H˘pital.
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  10. #10
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    Re: limit as x -> infty

    Quote Originally Posted by MarkFL2 View Post
    Right, on the inside of the log function, you now have the correct form to use the rule bought by L'H˘pital.
    I'm assuming the same thing applies to trig functions if there were a similar case
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  11. #11
    MHF Contributor MarkFL's Avatar
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    Re: limit as x -> infty

    Yes, I believe so as long as the trig function is continuous.
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