Příklad 2.5

Spočtěte limitu \(\large\lim_{x\to0^+}\sqrt{x}\ln(x)\)

$$\Large \begin{aligned}

& \lim_{x\to0^+}\sqrt{x}\ln(x) \

=& \lim_{x\to0^+}\frac{\ln(x)}{\frac{1}{\sqrt{x}}} \ =& \lim_{x\to0^+}-\frac{\infty}{\infty} \

\\ &\text{It's L'Hospital Time:} \\

=& \lim_{x\to0^+} \frac{(\ln(x))^{'}}{\left(\frac{1}{\sqrt{x}}\right)^{'}} \ =& \lim_{x\to0^+} \frac{\frac{1}{x}}{-\frac{1}{2}x^{-\frac{3}{2}}} \ =& \lim_{x\to0^+} \frac{1}{x}\cdot\frac{-2}{x^{-\frac{3}{2}}} \ =& \lim_{x\to0^+} \frac{1}{x}\cdot(-2x^{\frac{3}{2}}) \ =& \lim_{x\to0^+} \frac{1}{x}\cdot(-2\cdot{x^1}\cdot{}x^{\frac{1}{2}}) \ =& \lim_{x\to0^+} \frac{1}{\cancel{x}^1}\cdot(-2\cdot\cancel{{x^1}}^1\cdot{}x^{\frac{1}{2}}) \ =& \lim_{x\to0^+} (-2x^{\frac{1}{2}}) \ =& \lim_{x\to0^+} -2 \cdot \lim_{x\to0^+} x^{\frac{1}{2}} \ =& -2 \cdot 0 \ =& \boxed{0}

\end{aligned} \ $$