Question

If $\underset{x\to a}{lim}\frac{x^9-a^9}{x-a}=9$ , find all possible values of $a$.

Answer 1

$\underset{x\to a}{lim}\frac{x^9-a^9}{x-a}=\frac{0}{0}$ which is the indeterminate form

Therefore, L-hospitals rule is applied. i.e.,we take the derivative of numerator and denominator

$\underset{x\to a}{lim}\frac{\frac{d}{dx}(x^9-a^9)}{\frac{d}{dx}(x-a)}=\underset{x\to a}{lim}9x^8=9a^8$

Given that $\underset{x\to a}{lim}\frac{x^9-a^9}{x-a}=9$

Therefore, $9a^8=9$

$⟹a^8=1$

$⟹a=1$