Question

Find the limit (if it exists). If it does not exist, explain why. $$ \lim _{x \rightarrow \pi} \cot x $$

Short answer

The limit of \( \cot x \) as \( x \) approaches \( \pi \) does not exist because division by zero is undefined in math.

Step-by-step solution

Identifying the function

First identify the function that we are working with. In this problem the function is \( \cot x \). The cotangent function is defined as the reciprocal of the tangent function, that is, \( \cot x =\frac{1}{\tan x} \). Since \( \tan x \) is not defined for some values of \( x \) like \( \frac{\pi}{2} \) and \( \frac{3\pi}{2} \), the same will apply to \( \cot x \). Also, remember that \( \tan x = \frac{\sin x}{\cos x} \). So, another way of representing the cotangent function is: \( \cot x = \frac{\cos x}{\sin x} \).

Evaluate the cotangent function as \( x \) approaches \( \pi \)

Here the limit \( x \) is approaching \( \pi \). In this step, we plug \( \pi \) into the \( \cot x \) function. This is equivalent to trying to find \( \cot(\pi) \). From the unit circle, we know that \( \sin(\pi) = 0 \) and \( \cos(\pi) = -1 \). Therefore, \( \cot(\pi) = \frac{-1}{0} = undefined \).

Provide interpretation

The limit of \( \cot x \) as \( x \) approaches \( \pi \) is undefined because division by zero is undefined in math. Thus, the limit as \( x \) approaches \( \pi \) does not exist.