Answer
$y=\frac{x+2}{\cos x}$ is continuous for all $x\ne\frac{\pi}{2}+k\pi(k\in Z)$
Work Step by Step
$$y=\frac{x+2}{\cos x}$$
- Domain: $y$ is defined where $\cos x\ne0$, which means $x\ne\frac{\pi}{2}+k\pi (k\in Z)$
- As $x$ approaches any values of $\frac{\pi}{2}+k\pi(k\in Z)$, $\frac{x+2}{\cos x}$ approaches infinity, and it does reach for any definite value.
In other words, $\lim_{x\to(\pi/2+k\pi)}\frac{x+2}{\cos x}$ does not exist, so the function is discontinuous at all points $x=\frac{\pi}{2}+k\pi$.
So for all $x\ne\frac{\pi}{2}+k\pi$:
- $\lim_{x\to c}\cos x=\cos c$, so $y=\cos x$ is continuous in the domain defined.
- $\lim_{x\to c} (x+2)=c+2$, so $y=x+2$ is also continuous in the domain defined.
According to Theorem 8, the division of 2 functions continuous at $x=c$ is also continuous at $x=c$ (except when the denominator equals $0$)
Therefore, $y=\frac{x+2}{\cos x}$ is continuous for all $x\ne\frac{\pi}{2}+k\pi(k\in Z)$
