Question

at what points is the function ( y=\frac{cos x}{3x} ) continuous?
describe the set of ( x )-values where the function is continuous, using interval notation.
(simplify your answer. type your answer in interval notation.)

Explanation:

Step1: Analyze the domain of the function

The function \( y=\frac{\cos x}{3x} \) is a quotient of two functions. The cosine function \( \cos x \) is continuous for all real numbers \( x\in(-\infty,\infty) \). For the quotient \( \frac{f(x)}{g(x)} \) (here \( f(x)=\cos x \), \( g(x) = 3x \)), the function is undefined when \( g(x)=0 \).
Set \( 3x = 0 \), then \( x = 0 \).

Step2: Determine the continuity intervals

Since the function \( y=\frac{\cos x}{3x} \) is a quotient of continuous functions (a trigonometric function \( \cos x \) and a polynomial function \( 3x \)) and is undefined only at \( x = 0 \), by the quotient - rule of continuity (if \( f(x) \) and \( g(x) \) are continuous on an interval \( I \) and \( g(x)
eq0 \) on \( I \), then \( \frac{f(x)}{g(x)} \) is continuous on \( I \)), the function \( y=\frac{\cos x}{3x} \) is continuous on \( (-\infty,0)\cup(0,\infty) \).

Answer:

\((-\infty,0)\cup(0,\infty)\)