Question

Find all the possible values of $a$, if $\underset{x\to a}{lim}\frac{x^3-x^3}{x-a}=\underset{x\to 1}{lim}\frac{x^4-1}{x-1}.$

Answer 1

$\underset{x\to 0}{lim}\frac{x^3-a^3}{x-a}=\underset{x\to 0}{lim}\frac{x^4-1}{x-1}$

$\Rightarrow 3{\left(a\right)}^2=4{\left(1\right)}^3\left[\sin ce\underset{x\to a}{lim}\frac{x^n-a^n}{x-a}=n{a}^{n-1}\right]$

$\Rightarrow 3a^2=4$

$\Rightarrow a^2=\frac{4}{3}$

$\Rightarrow a=\pm \frac{2}{\sqrt{3}}$