Question

$\lim_{x \to \0} \frac{sin3x-sinx}{ln(1+x)}$

Answer 1

$\lim\limits _{x \to 0} \frac{sin3x-sinx}{ln(1+x)} = \lim\limits _{x \to 0} \frac{2sinx\cdot cos2x}{ln(1+x)} =[\; sinx\sim x\; ,\; \; ln(1+x)\sim x\; ]=\\\\= \lim\limits _{x \to 0} \frac{2x\cdot cos2x}{x}=\lim\limits _{x\to 0} (2cos2x)=2\cdot cos0=2\cdot 1=2$