\(\lim\limits_{x \to 1}\frac{x^{15}-1}{x^{10}-1}\)
= \(\lim\limits_{x \to 1}\frac{(x^5)^3-1^3}{(x^5)^2-1^2}\)
∵ a3 − b3 = (a − b) (a2 + b2 + ab)
\(=\lim\limits_{x \to 1}\frac{(x^5-1)(x^{10}+x^5+1)}{(x^5-1)(x^5+1)}\)
= \(\lim\limits_{x \to 1}\frac{x^{10}+x^5+1}{x^5+1}\)
= \(\frac{1^{10}+1^{5}+1}{1^5+1}\)
= \(\frac{3}{2}\)
