Question

Find the limit if exist
limit x->0 sin4x/sin6x

Answer 1

Use l'hopsital's since sin(0)/sin(0)=0/0 indeterminate form so:
\[\lim_{x \rightarrow 0} \frac{\frac{d}{dx}\sin(4x)}{\frac{d}{dx}\sin(6x)}=\lim_{x \rightarrow 0} \frac{4 \cos(4x)}{6 \cos(6x)}=\frac{4}{6}=\frac{2}{3}\]
Verification:
http://www.wolframalpha.com/input/?i=limit+x-%3E0+sin4x%2Fsin6x