Calculate $\underset{x}{lim}\frac{ln(5x)}{\sqrt[5]{x}}.$

Solution

Since $ln(5x)$ and $asx,$ l'Hospital's Rule applies.

$\underset{x}{lim}\frac{ln(5x)}{\sqrt[5]{x}}=\underset{x}{lim}\frac{\frac{1}{x}}{\frac{1}{5}{x}^{-\frac{4}{5}}}=\underset{x}{lim}\frac{\frac{1}{x}}{\frac{1}{(\sqrt[5]{x^4})}}$

$\underset{x}{lim}\frac{ln(5x)}{\sqrt[5]{x}}=\underset{x}{lim}\frac{\frac{1}{x}}{\frac{1}{(\sqrt[5]{x^4})}}=\underset{x}{lim}\frac{5}{\sqrt[5]{x}}=$

$.$