Consider a function $f: \mathbf{R} \rightarrow \mathbf{R}$ such that $f(x)=1$ if $x$ is rational, and $f(x)=1-\epsilon,$ where $0<\epsilon<1$, if $x$ is irrational. Which of the following is $\text{TRUE}?$
- $\lim _{x \rightarrow \infty} f(x)=1$
- $\lim _{x \rightarrow \infty} f(x)=1-\epsilon$
- $\lim _{x \rightarrow \infty} f(x)$ exists, but is neither 1 nor $1-\epsilon$
- $\max _{x \geq 1} f(x)=1$
- None of the above