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How to evaluate $$\lim\limits_{x\to1}\frac{x^{m+1}-x^{n+1}+x^n-mx+m-1}{(x-1)^2}$$

I used substitution $t=x-1 \Rightarrow t\rightarrow 0$

After this, the limit is:

$$\lim\limits_{t\to0}\frac{(t+1)^{m+1}-(t+1)^{n+1}+(t+1)^n-m(t+1)+m-1}{t^2},m,n \in \mathbb{N}$$

Then I used L'Hospital's rule, but I get $\frac{const.}{0}$

Can L'Hospital's rule be used, or some other method?

Thanks for replies.

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    $\begingroup$ Hint: $a^n-b^n=(a-b)(a^{n-1}+ba^{n-2}+...+b^{n-2}a+b^{n-1})$. $\endgroup$ May 23 '15 at 15:21
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I think there is no truck here, it suffices to apply l'Hospital's rule twice only \begin{eqnarray*} \lim_{x\rightarrow 1}\frac{x^{m+1}-x^{n+1}+x^{n}-mx+m-1}{(x-1)^{2}} &=&\lim_{x\rightarrow 1}\frac{(m+1)x^{m}-(n+1)x^{n}+nx^{n-1}-m}{2(x-1)}\ (by\ HRule:\frac{0}{0}) \\ &=&\lim_{x\rightarrow 1}\frac{m(m+1)x^{m-1}-n(n+1)x^{n-1}+n(n-1)x^{n-2}}{2}\ \ (by\ HRule:\frac{0}{0}) \\ &=&\frac{m(m+1)-n(n+1)+n(n-1)}{2} \\ &=&\frac{m(m+1)-n^{2}-n+n^{2}-n}{2} \\ &=&\frac{m(m+1)}{2}-n. \end{eqnarray*}

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If you're getting $c/0$ when using L'Hospital's Rule, then the limit doesn't exist.

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Hint: Use L'Hospital's rule twice, and you will get 2 in the denominator and be able to evaluate the limit. (After the first application I believe you will still get $0/0$.)

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The numerator can be written $$ x^{m+1}-1-x^{n+1}+x^n-mx+m= (x-1)\biggl(\biggl(\sum_{k=0}^m x^k\biggr)-x^n-m\biggr) $$ Set $f_m(x)=\sum_{k=0}^m x^k$, so your limit can be written as $$ \lim_{x\to1}\frac{f_m(x)-x^n-m}{x-1} $$ Note that $f_m(1)=m+1$, so we can apply l'Hôpital, giving $$ \lim_{x\to1}(f_m'(x)-nx^{n-1}) $$ Since $$ f_m'(x)=\sum_{k=1}^m kx^{k-1} $$ we have $$ f_m'(1)=1+2+\dots+m=\dots $$ and it should be clear how to finish.

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If you use the binomial expansion or Taylor series, the numerator is something like $$\frac{1}{2} \left(m^2+m-2 n\right)(x-1)^2+\cdots$$

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  • $\begingroup$ How did you get the expression in numerator? $\endgroup$
    – user300045
    May 23 '15 at 15:55
  • $\begingroup$ As I said, binomial expansion. Isn't correct ? $\endgroup$ May 23 '15 at 16:25

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