Zadatak 0027

Test: MPsIVKVII_005

 

Izračunaj: \(\lim_{x \to 0}\left ( \frac{1}{x}-\frac{1}{e^{x}-1} \right )\).

 

Rešenje:

 

\(...=\lim_{x \to 0}\frac{e^{x}-1-x}{x\left ( e^{x}-1 \right )}\) \(=\left ( \frac{0}{0} \right )=\lim_{x \to 0}\frac{e^{x}-1}{e^{x}-1+xe^{x}}\) \(=\left ( \frac{0}{0} \right )=\lim_{x \to 0}\frac{e^{x}}{e^{x}+e^{x}+xe^{x}}\) 

 

\(=\lim_{x \to 0}\frac{e^{x}}{e^{x}(2+x)}=\frac{1}{2}\)