What's the limit of $\lim_{x\to -1} \frac{x^{101}+1}{x+1} $

Question

I've tried a lot in how to solve this limit, What's is the limit of $$\lim_{x\to -1} \frac{x^{101}+1}{x+1} $$

Answer 1

I go with mercio (see the comments above) which seems to be the most elegant way. Put $f(x) = x^{101}$. Then $f'(x) = 101x^{100}$ and hence $$ \lim_{x\to -1}\frac{x^{101}+1}{x+1} = \lim_{x\to -1}\frac{f(x)-f(-1)}{x-(-1)} = f'(-1) = 101. $$

Answer 2

$$x^{101}+1=(x+1)(x^{100}-x^{99}+x^{98}\cdots+x^2-x+1)$$ $$\lim_{x\to-1}\frac{x^{101}+1}{x+1}=\sum_{k=0}^{100} (-1)^{2k}=101$$