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Calculating Limit

  1. Sep 24, 2015 #1
    1. The problem statement, all variables and given/known data
    $$\lim _{x \rightarrow 1} (\frac{23}{1-x^{23}}-\frac{11}{1-x^{11}})$$


    2. Relevant equations
    i) For functions f and g which are differentiable on an open interval I except possibly at a point c contained in I, if
    2831c94ace338a268ca7cee5a6c2dd68.png , and
    09d577aee808027079cf3191c0800309.png exists, and
    21cc02743caf32a7473a553a60deaeb8.png for all x in I with xc,

    then

    [PLAIN]https://upload.wikimedia.org/math/8/9/9/8991dfbd9db5990224ae803c727464a7.png. [Broken]

    ii) $$\lim _{x \rightarrow a} (f(x) \cdot g(x)) = \lim _{x \rightarrow a} f(x) \cdot \lim _{x \rightarrow a} g(x) $$
    3. The attempt at a solution
    $$\begin{align}
    \lim _{x \rightarrow 1} (\frac{23}{1-x^{23}}-\frac{11}{1-x^{11}})&= \lim _{x \rightarrow 1} \frac{1}{1-x}(\frac{23(1-x)}{1-x^{23}}-\frac{11(1-x)}{1-x^{11}})\\
    &= \lim _{x \rightarrow 1} \frac{1}{1-x} \cdot (\lim _{x \rightarrow 1} \frac{23(1-x)}{1-x^{23}}-\lim _{x \rightarrow 1} \frac{11(1-x)}{1-x^{11}})\\
    &= \lim _{x \rightarrow 1} \frac{1}{1-x} \cdot (\lim _{x \rightarrow 1} \frac{1}{x^{22}}-\lim _{x \rightarrow 1} \frac{1}{x^{10}}) \text {[Using L Hopital's Rule]} \\
    &= \lim _{x \rightarrow 1} \frac{1-x^{12}}{1-x} \lim _{x \rightarrow 1} x^{22} \\
    &= \lim _{x \rightarrow 1} 12x^{11} \lim _{x \rightarrow 1} x^{22} \text {[Using L Hopital's Rule]}\\
    &= 12
    \end{align}$$
    But the correct answer is 6
     
    Last edited by a moderator: May 7, 2017
  2. jcsd
  3. Sep 24, 2015 #2

    BvU

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    Apparently somehow you are subtracting two infinities. I am used to letting Taylor series do the work for limits, but that doesnt fly here: $$\lim_{\epsilon\downarrow 0} {23\over 1-(1+\epsilon)^{23} } = \lim_{\epsilon\downarrow 0} {23\over 23\epsilon}$$ and there you go. Same with the other one. Conclusion: one more Taylor term needed and then you'll get the ##{1\over2}## to produce the book result.
     
  4. Sep 24, 2015 #3

    Ray Vickson

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    You cannot write
    [tex] \lim_{x \to 1} \frac{1}{1-x} \cdot \left( \frac{23(1-x)}{1-x^{23}} - \frac{11(1-x)}{1-x^{11}} \right) [/tex]
    as
    [tex] \lim_{x \to 1} \frac{1}{1-x} \cdot \left( \lim_{x \to 1} \frac{23(1-x)}{1-x^{23}} - \lim_{x \to 1} \frac{11(1-x)}{1-x^{11}} \right) [/tex]
    because the first factor ##= \pm \infty##. Instead, take ##x = 1 + h## and expand out ##x^n = (1+h)^n## in powers of ##h##.
     
    Last edited by a moderator: May 7, 2017
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