[SPLIT] find lim x ->0+ of f(x) = sin(x) ln(x)

[mathhelp]

find the limit x-->0+ of f(x) = sin(x) ln(x)

thanks

 

[skeeter]

\(\displaystyle \L \lim_{x \rightarrow 0^+} \sin{x} \ln{x} =\)

\(\displaystyle \L -\lim_{x \rightarrow 0^+} \frac{-\ln{x}}{\csc{x}}\)

as x -> 0⁺, both numerator and denominator approach infinity ... use L'Hopital's rule.

\(\displaystyle \L -\lim_{x \rightarrow 0^+} \frac{-\frac{1}{x}}{-\csc{x}\cot{x}}=\)

\(\displaystyle \L -\lim_{x \rightarrow 0^+} \frac{\sin{x}\tan{x}}{x}=\)

\(\displaystyle \L -\lim_{x \rightarrow 0^+} \frac{\sin{x}}{x}\tan{x}= -1 \cdot 0 = 0\)