Question

$\underset{x\to a}{lim}\frac{x^{2/3}-a^{2/3}}{x^{3/4}-a^{3/4}}$

Answer 1

$L={lim}_{x\to a}\frac{x^{\frac{2}{3}}-a^{\frac{2}{3}}}{x^{\frac{3}{4}}-a^{\frac{3}{4}}}$

we have ${lim}_{x\to \infty }\frac{x^n-a^n}{x-a}=n{a}^{n-1}$

multiplying denominator and numerator by $(x-a)$ ,we have

$L={lim}_{x\to a}\frac{(x^{\frac{2}{3}}-a^{\frac{2}{3}})}{(x-a)}\times \frac{(x-a)}{(x^{\frac{3}{4}}-a^{\frac{3}{4}})}={lim}_{x\to a}\left(\frac{x^{\frac{2}{3}}-a^{\frac{2}{3}}}{x-a}\right)\times {lim}_{x\to a}\left(\frac{x-a}{x^{\frac{3}{4}}-a^{\frac{3}{4}}}\right)=\frac{2}{3}{a}^{\frac{2}{3}-1}\times \frac{1}{\frac{3}{4}{a}^{\frac{3}{4}-1}}=\frac{2}{3}\times \frac{1}{\frac{3}{4}}\times \frac{a^{\frac{-1}{3}}}{a^{\frac{-1}{4}}}=\frac{8}{9}{a}^{\frac{-1}{3}+\frac{1}{4}}=8a^{\frac{-1}{12}}$