Chapter 1.3, Problem 8E

To determine

The points of discontinuity of the function. Identify the type of

Discontinuity.

Answer to Problem 8E

The function is

Explanation of Solution

Given:

  $y=\cot x$

Calculation:

The given equation $y=\cot x$

Rewrite it as:

  $y=\frac{\cos x}{\sin x}$

Here, both the numerator and denominator are continuous functions. So, the function

  $y=\cot x$ is continuous.

Find the discontinuity point of the function$y=\cot x$:

Set the argument in $\cot \left(x\right)$ equal to $\pi n$ , to find where the expression is undefined:

  $x=\pi n$ for any integer $n$

The equation is undefined where the denominator equals $0$ , the argument of a square

root is less than $0$ , or the argument of a logarithm is less than or equal to $0$ .

  $\left\{x|x=\pi n\right\}n$ , for any integer $n$

Thus, the function is continuous and have discontinuity at $x=\pi n$ .