lim_{x\\to0} x\\cot x

Explanation:

Step1: Rewrite cotangent

Recall that $\cot x=\frac{\cos x}{\sin x}$, so the limit becomes $\lim_{x\rightarrow0}x\cdot\frac{\cos x}{\sin x}=\lim_{x\rightarrow0}\frac{x}{\sin x}\cdot\cos x$.

Step2: Use known limit

We know that $\lim_{x\rightarrow0}\frac{\sin x}{x} = 1$, so $\lim_{x\rightarrow0}\frac{x}{\sin x}=1$. Also, $\lim_{x\rightarrow0}\cos x=\cos(0) = 1$.

Step3: Apply limit - product rule

By the product - rule for limits $\lim_{x\rightarrow a}(f(x)\cdot g(x))=\lim_{x\rightarrow a}f(x)\cdot\lim_{x\rightarrow a}g(x)$, we have $\lim_{x\rightarrow0}\frac{x}{\sin x}\cdot\cos x=\lim_{x\rightarrow0}\frac{x}{\sin x}\cdot\lim_{x\rightarrow0}\cos x$.

Answer:

$1$