Question Number 192846 by mathlove last updated on 29/May/23
$$\underset{{h}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{3}{h}}{\:\sqrt[{\mathrm{5}}]{\mathrm{3}{h}+{x}}−\sqrt[{\mathrm{5}}]{{x}}}=? \\ $$
Answered by MM42 last updated on 29/May/23
$${for}\:\:{f}\left({x}\right)=\sqrt[{\mathrm{5}}]{{x}}\:\Rightarrow{f}'\left({x}\right)=\frac{\mathrm{1}}{\mathrm{5}\sqrt[{\mathrm{5}}]{{x}^{\mathrm{4}} }} \\ $$$${lim}_{{h}\rightarrow\mathrm{0}} \frac{\mathrm{3}{h}}{\:\sqrt[{\mathrm{5}}]{\mathrm{3}{h}+{x}}−\sqrt[{\mathrm{5}}]{{x}}}=\frac{\mathrm{1}}{{f}'\left({x}\right)}=\mathrm{5}\sqrt[{\mathrm{5}}]{{x}^{\mathrm{4}} } \\ $$