[tex] \text{ln}x = t \Rightarrow x = e^t \\ \\ x \to 0 \Rightarrow e^t \to 0 = t \to -\infty\\ \\ \Rightarrow \underset{x>0}{ \lim\limits_{x \to 0}}\dfrac{\text{ln}x}{x} = \lim\limits_{t \to -\infty} \dfrac{t}{e^t} = \\ \\ = \dfrac{-\infty}{e^{-\infty}} = \dfrac{-\infty}{0_+} = -\infty[/tex]
