Question

evaluate limit as x approaches zero from the right. sin (x) ln (3x).

Answer 1

\[\lim_{x \rightarrow 0^+} \frac{\ln(3x)}{\csc(x)}=\lim_{x \rightarrow 0^+}\frac{\frac{1}{x}}{-\cot^2(x)}\]
\[\lim_{x \rightarrow 0^+}\frac{1}{-xcot^2(x)}=\lim_{x \rightarrow 0^+}\frac{\sin^2(x)}{x \cos^2(x)}\]
\[=\lim_{x \rightarrow 0^+}\frac{\sin(x)}{x} \sin(x) \frac{1}{\cos^2(x)} =(1)(0)(\frac{1}{1^2})=0\]