<s>[INST] Calculating $\lim_{x \rightarrow 1}(\frac{23}{1-x^{23}} - \frac{31}{1-x^{31}})$ How to calculate following limit?
$$\lim_{x \rightarrow 1}\left(\frac{23}{1-x^{23}} - \frac{31}{1-x^{31}}\right)$$
 [/INST] Generalized form
$$\lim_{x \rightarrow 1}\left(\frac{m}{1-x^{m}} - \frac{n}{1-x^{n}}\right) = \frac{n-m}{2}$$ 
$Proof:$
$Let$ $L =   \lim_{x \rightarrow 1}\left(\frac{m}{1-x^{m}} - \frac{n}{1-x^{n}}\right)$
$Let x = \frac{1}{y}$
$if x\rightarrow 1$     $then$ $y\rightarrow 1$
$L =\lim_{x \rightarrow 1}\left(\frac{m}{1-\frac{1}{y^{m}}} - \frac{n}{1-\frac{1}{y^{n}}}\right) $
$L =\lim_{x \rightarrow 1}\left(\frac{my^{m}}{y^{m}-1} - \frac{ny^{n}}{y^{n}-1}\right)$
$L =\lim_{x \rightarrow 1}\left(\frac{my^{m}-m+m}{y^{m}-1} - \frac{ny^{n}-n+n}{y^{n}-1}\right)$
$L =\lim_{x \rightarrow 1}\left(\frac{(-m)(1-y^{m})+m}{y^{m}-1} - \frac{(-n)(1-y^{n})+n}{1-y^{n}}\right)$
$L =\lim_{x \rightarrow 1}\left((-m)+\frac{m}{y^{m}-1} - (-n)-\frac{n}{y^{n}-1}\right)$
$L =\lim_{x \rightarrow 1}\left((n-m)+\frac{m}{y^{m}-1} - \frac{n}{y^{n}-1}\right)$
$L =\lim_{x \rightarrow 1}(n-m)+\lim_{x \rightarrow 1}\left(-\frac{m}{1-y^{m}} + \frac{n}{1-y^{n}}\right)$
$L =(n-m)-\lim_{x \rightarrow 1}\left(\frac{m}{1-y^{m}} - \frac{n}{1-y^{n}}\right)$
$L =(n-m)-L$
$2L =(n-m)$
$L =\frac{(n-m)}{2}$
For above question m = 31,  n =  23
hence $L= \frac{(23-31)}{2} = (-4)$  
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