Oblicz granicę funkcji

[wojtasekpl]

\(\Lim_{x\to 0} \frac{4-4cos3x}{ln(x+1)-x} \)

\(\Lim_{x\to 0} \frac{1}{ln(x+1)} - \frac{1}{x} \)

[eresh]

\(\Lim_{x\to 0} \frac{4-4cos3x}{ln(x+1)-x} \)

\(\Lim_{x\to 0}\frac{4-4\cos 3x}{\ln (x+1)-x}=_H\Lim_{x\to 0}\frac{12\sin 3x}{\frac{1}{x+1}-1}=_H\Lim_{x\to 0}\frac{36\cos 3x}{-\frac{1}{(x+1)^2}}=-36\)

[eresh]

\(\Lim_{x\to 0} \frac{1}{ln(x+1)} - \frac{1}{x} \)

\(\Lim_{x\to 0}\frac{x-\ln (x+1)}{x\ln (x+1)}=_H\Lim_{x\to 0}\frac{1-\frac{1}{x+1}}{\ln (x+1)+x\cdot\frac{1}{x+1}}=_H\Lim_{x\to 0}\frac{\frac{1}{(x+1)^2}}{\frac{1}{x+1}+\frac{x+1-x}{(x+1)^2}}=\frac{1}{2}\)