다음 극한값을 계산하라.
$$\lim_{x\to1}\frac{x^{2008}-1}{x-1}$$
풀이
\(\displaystyle\lim_{x\to1}\frac{x^{2008}-1}{x-1}=\lim_{x\to1}\frac{(x-1)(x^{2007}+x^{2006}+\cdots+x^2+x^1+1)}{x-1}\)
\(\displaystyle=\lim_{x\to1}(x^{2007}+x^{2006}+\cdots+x^2+x^1+1)\)
\(\displaystyle=1^{2007}+1^{2006}+\cdots+1^2+1^1+1=2008\)