Question

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

Answer 1

At $x\to 1$,

$\frac{x^{15}-1}{x^{10}-1}$ becomes $\frac{0}{0}$ form

Now,
$\underset{x\to 1}{lim}\frac{x^{15}-1}{x^{10}-1}$

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

Divide and multiply by $(x-1)$, we get
$=\underset{x\to 1}{lim}[\frac{x^{15}-1^{15}}{x-1}\times \frac{x-1}{x^{10}-1^{10}}]$

$=\frac{\underset{x\to 1}{lim}\left[\frac{x^{15}-1^{15}}{x-1}\right]}{\underset{x\to 1}{lim}\left[\frac{x^{10}-1^{10}}{x-1}\right]}$

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

$=\frac{3}{2}$