Question

Evaluate: $\underset{x\to 1}{lim}\frac{x^{15}-1}{x^{10}-1}$

Answer 1

We have,

$\underset{x\to 1}{lim}\frac{x^{15}-1}{x^{10}-1}$
Dividing in the numerator and denominator by $(x-1)$, we get
$\underset{x\to 1}{lim}\left(\frac{x^{15}-1^{15}}{x-1}\right)\left(\underset{x\to 1}{lim}\frac{1}{\frac{(x^{10}-1^{10})}{(x-1)}}\right)$

$=\frac{15(1{)}^{14}}{10(1{)}^9}$ [Since, $\underset{x\to a}{lim}\frac{x^n-a^n}{x-a}=n{a}^{n-1}$]

$=\frac{3}{2}$