Answer
See explanations.
Work Step by Step
Step 1. To prove the limit $\lim_{x\to\infty}f(x)=k$, for any small value $\epsilon\gt0$, there is a corresponding value $M$ so that for all $x\gt M$, we get $|f(x)-k|\lt\epsilon$
Step 2. As this function has a constant value $f(x)=k$, for for any small value $\epsilon\gt0$, we can choose $M=1$ so that for all $x\gt M$, we have $|f(x)-k|=|k-k|=0\lt\epsilon$ which proves the limit statement.