Question

at what points is the function $y = \frac{\sin x}{5x - 5}$ 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: Find where the denominator is zero

To determine the points of discontinuity, we first find when the denominator \(5x - 5 = 0\). Solving for \(x\):
\[

$$\begin{align*} 5x - 5 &= 0\\ 5x &= 5\\ x &= 1 \end{align*}$$

\]

Step2: Determine the domain of continuity

The function \(y=\frac{\sin x}{5x - 5}\) is a rational function (a quotient of two functions, where the numerator \(\sin x\) is continuous everywhere and the denominator is a polynomial). A rational function is continuous everywhere except where the denominator is zero. So the function is continuous for all real numbers except \(x = 1\). In interval notation, this is \((-\infty, 1)\cup(1, \infty)\).

Answer:

\((-\infty, 1)\cup(1, \infty)\)