Question

If $\underset{x\to a}{lim}\frac{x^5-a^5}{x-a}=405$, find all possible values of a

Answer 1

$\underset{x\to a}{lim}\frac{x^5-a^5}{x-a}=\frac{0}{0}$ which is the indeterminate form

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

$\underset{x\to a}{lim}\frac{\frac{d}{dx}(x^5-a^5)}{\frac{d}{dx}(x-a)}=\underset{x\to a}{lim}5x^4=5a^4$

Given that $\underset{x\to a}{lim}\frac{x^5-a^5}{x-a}=405$

Therefore, $5a^4=405$

$⟹a^4=81$

$⟹a^4=3^4$

$⟹a=\pm 3,\pm 3i$