Answer
\[\lim_{x\rightarrow 0}(\csc x-\cot x)=0\]
Work Step by Step
Let \[l=\lim_{x\rightarrow 0}(\csc x-\cot x)\]
Which is $\infty-\infty$ form
\[l=\lim_{x\rightarrow 0}\left(\frac{1}{\sin x}-\frac{\cos x}{\sin x}\right)\]
\[l=\lim_{x\rightarrow 0}\left(\frac{1-\cos x}{\sin x}\right)\]
Which is $\frac{0}{0}$ form
Using L' Hopital's rule
\[l=\lim_{x\rightarrow 0}\frac{(1-\cos x)'}{(\sin x)'}\]
\[l=\lim_{x\rightarrow 0}\frac{\sin x}{\cos x}\]
\[\Rightarrow l=\frac{0}{1}=0\]
Hence,
\[\lim_{x\rightarrow 0}(\csc x-\cot x)=0\]
