Evaluate :\(\lim\limits_{x \to 1}\frac{x^{15}-1}{x^{10}-1}\)

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

Answer»

\(\lim\limits_{x \to 1}\frac{x^{15}-1}{x^{10}-1}\)

= \(\lim\limits_{x \to 1}\frac{(x^5)^3-1^3}{(x^5)^2-1^2}\)

∵ a³ − b³ = (a − b) (a²+ b² + ab)

\(=\lim\limits_{x \to 1}\frac{(x^5-1)(x^{10}+x^5+1)}{(x^5-1)(x^5+1)}\)

= \(\lim\limits_{x \to 1}\frac{x^{10}+x^5+1}{x^5+1}\)

= \(\frac{1^{10}+1^{5}+1}{1^5+1}\)

= \(\frac{3}{2}\)

Discussion

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