# Calculating Limit

1. Sep 24, 2015

### arpon

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
, and
exists, and
for all x in I with xc,

then

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. Sep 24, 2015

### BvU

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.

3. Sep 24, 2015

### Ray Vickson

You cannot write
$$\lim_{x \to 1} \frac{1}{1-x} \cdot \left( \frac{23(1-x)}{1-x^{23}} - \frac{11(1-x)}{1-x^{11}} \right)$$
as
$$\lim_{x \to 1} \frac{1}{1-x} \cdot \left( \lim_{x \to 1} \frac{23(1-x)}{1-x^{23}} - \lim_{x \to 1} \frac{11(1-x)}{1-x^{11}} \right)$$
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