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Math Help - Limit of a composite function

  1. #1
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    Limit of a composite function

    I don't want an easy answer to this problem. However, I would be happy if you could provide me with theorems and/or techniques required to solve it.


    $\lim _{x\to \infty }x^2(ln({x+1\over x}) +ln({2x+3\over 2x}))$

    I know that $\lim _{x\to \infty }({x+1\over x})^x = \lim _{x\to \infty }(1+{1\over x})^x = e $ But the natural logarithm is in the way and I think that you can't calculate the limit IN the logarithm first.

    Thanks in advance

    P.S. Oh, and by the way could anyone recommend a book on calculus with challenging problems, because most of the usual calculus textbooks aren't rigorous enough for my college course. (My lecturer always finds a way to give much more complicated problems than those in textbooks)
    Last edited by Doubled144314; December 3rd 2012 at 02:19 AM.
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  2. #2
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    Re: Limit of a composite function

    Hello, Doubled144314!

    We need these two theorems:

    . . \lim_{x\to\infty}\left(1 + \tfrac{1}{x}\right)^x \;=\;e

    . . \lim_{x\to\infty}\left(1 + \tfrac{a}{x}\right)^x \;=\;e^a


    \displaystyle \lim _{x\to\infty}x^2\bigg[\ln\left(\tfrac{x+1}{x}\right) +\ln\left(\tfrac{2x+3}{2x}\right)\bigg]

    We have: . \lim_{x\to\infty}x\cdot x\bigg[\ln\left(1+\tfrac{1}{x}\right) + \ln\left(1 + \tfrac{3}{2x}\right)\bigg]

    . . . . . . =\;\lim_{x\to\infty}x\bigg[x\ln\left(1+\tfrac{1}{x}\right) + x\ln\left(1 + \tfrac{3}{2x}\right)^x\bigg]

    . . . . . . =\;\lim_{x\to\infty}x\bigg[\ln\left(1 + \tfrac{1}{x}\right)^x + \ln\left(1 + \tfrac{\frac{3}{2}}{x}\right)^x\bigg]

    . . . . . . =\;\lim_{x\to\infty}x\cdot \bigg[\ln\left(\lim_{x\to\infty}\left[1 + \tfrac{1}{x}\right]^x\right) + \ln\left(\lim_{x\to\infty}\left[1 + \tfrac{\frac{3}{2}}{x}\right]^x\right)\bigg]

    . . . . . . =\;\infty\cdot \ln(e)\cdot \ln(e^{\frac{3}{2}}) \;=\;\infty\cdot1\cdot\tfrac{3}{2} \;=\;\infty
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  3. #3
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    Re: Limit of a composite function

    Quote Originally Posted by Doubled144314 View Post
    I don't want an easy answer to this problem. However, I would be happy if you could provide me with theorems and/or techniques required to solve it.


    $\lim _{x\to \infty }x^2(ln({x+1\over x}) +ln({2x+3\over 2x}))$

    I know that $\lim _{x\to \infty }({x+1\over x}) = \lim _{x\to \infty }(1+{1\over x})^1 = e $ But the natural logarithm is in the way and I think that you can't calculate the limit IN the logarithm first.

    Thanks in advance

    P.S. Oh, and by the way could anyone recommend a book on calculus with challenging problems, because most of the usual calculus textbooks aren't rigorous enough for my college course. (My lecturer always finds a way to give much more complicated problems than those in textbooks)
    I would rewrite the function as \displaystyle \begin{align*} \lim_{x \to \infty} \frac{\ln{\left(\frac{x+1}{x}\right)} + \ln{\left(\frac{2x+3}{2x}\right)}}{\frac{1}{x^2}} \end{align*}, and since this goes to \displaystyle \begin{align*} \frac{0}{0} \end{align*} you can apply L'Hospital's Rule.
    Last edited by Prove It; December 2nd 2012 at 07:05 PM.
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