Question

lim as x approaches 0 of lnx-lnsinx

Answer 1

\[\lim_{x \rightarrow 0} (lnx-lnsinx)\]

Answer 2

0

Answer 3

\[\lim_{x\to0}\left(\ln x-\ln\sin x\right)=\lim_{x\to0}\ln\left(\frac{x}{\sin x}\right)\]
I think you mean as x approaches 0 from the right...

In any case, since \(\ln x\) is defined for \(x>0\), you can apply the continuity principle:
\[\lim_{x\to0^+}\ln\left(\frac{x}{\sin x}\right)=\ln\left(\lim_{x\to0^+}\frac{x}{\sin x}\right)=\ln1\]

Answer 4

Interesting, according to the graph, \(\ln\dfrac{x}{\sin x}\) *is* defined for \(x<0\)... Disregard that small note I made.