[tex]\bf \lim\limits_{x\to 0}\ ln[sin(x)]-ln(x)\\\\
-----------------------------\\\\
log_{{ a}}\left( \frac{x}{y}\right)\implies log_{{ a}}(x)-log_{{ a}}(y)\qquad thus\\\\
-----------------------------\\\\
ln[sin(x)]-ln(x)\implies ln\left[ \cfrac{sin(x)}{x} \right]\implies ln(1)\implies 0[/tex]
