Answer
$$f(x)=\frac{x+6}{x+1}$$
Work Step by Step
$$f(x)=\frac{x+6}{x+1}$$
This function is continuous for all $x$ except $x=-1$, where it has a nonremovable discontinuity. The reason is stated as follows:
1) For all $x\ne-1$, we have $\lim_{x\to c}\frac{x+6}{x+1}=\frac{c+6}{c+1}=f(c)$, meaning that $f$ is continuous for all $x\ne-1$.
- As $x\to-1$, though, $(x+1)$ approaches $0$, so $\frac{x+6}{x+1}$ will approach $\infty$. This means $\lim_{x\to-1}f(x)$ does not exist. $f(x)$ is discontinuous at $x=-1$ as a result.
2) This discontinuity is a nonremovable one, because both $\lim_{x\to-1}f(x)$ does not exist. Therefore, there is no way we can extend $f(x)$ to include a value of $f(-1)=\lim_{x\to-1}f(x)$ in order to remove the discontinuity.
