Answer
$y=\frac{\cos x}{x}$ is continuous on $(-\infty,0)\cup(0,\infty)$
Work Step by Step
$$y=\frac{\cos x}{x}$$
- Domain: because $(-\infty,0)\cup(0,\infty)$
- As $x\to0$, $\frac{\cos x}{x}$ approaches infinity, and it does reach for any definite value.
In other words, $\lim_{x\to0}\frac{\cos x}{x}$ does not exist, so the function is discontinuous at $x=0$.
So for all $x\in(-\infty,0)\cup(0,\infty)$:
- $\lim_{x\to c}\cos x=\cos c$, so $y=\cos x$ is continous in the domain defined.
- $\lim_{x\to c} x=c$, so $y=x$ is also continous in the domain defined.
According to Theorem 8, the division of 2 continuous functions at $x=c$ is also continuous at $x=c$ (with $x\ne0$)
Therefore, $y=\frac{\cos x}{x}$ is continuous on $(-\infty,0)\cup(0,\infty)$
