evaluate the following limit. use lhôpitals rule when it is convenient and applicable.\n\\(lim_{x\rightarrowp…

Explanation:

Step1: Rewrite the function

First, rewrite ((x - \pi)\cot x) as (\frac{x-\pi}{\tan x}). As (x\rightarrow\pi), we have the indeterminate - form (\frac{0}{0}), so L'Hopital's Rule can be applied.

Step2: Differentiate the numerator and denominator

Differentiate the numerator (u=x - \pi), (u^\prime=1), and the denominator (v = \tan x), (v^\prime=\sec^{2}x) according to the derivative rules.

Step3: Find the limit of the new - function

By L'Hopital's Rule, (\lim_{x\rightarrow\pi}\frac{x - \pi}{\tan x}=\lim_{x\rightarrow\pi}\frac{1}{\sec^{2}x}).

Step4: Evaluate the limit

Since (\sec x=\frac{1}{\cos x}), when (x\rightarrow\pi), (\cos\pi=- 1), then (\sec\pi=-1) and (\sec^{2}x = 1). So (\lim_{x\rightarrow\pi}\frac{1}{\sec^{2}x}=1).

Answer:

1