Answer
$0$
Work Step by Step
Consider $f(x)=\lim\limits_{x \to 0} \csc x-\cot x=\lim\limits_{x \to 0} \dfrac{1}{\sin x}-\lim\limits_{x \to 0} \dfrac{\cos x}{\sin x}$
This can be further simplified as: $\lim\limits_{x \to 0} \dfrac{1}{\sin x}-\lim\limits_{x \to 0} \dfrac{\cos x}{\sin x}=\lim\limits_{x \to 0} \dfrac{(1-\cos x )\sin x}{(1-cos x)(1+\cos x)}$
Then $\lim\limits_{x \to 0}\dfrac{\sin x}{1+\cos x}=0$
