real analysis - How to use formal limit definition?

(a) IF $\lim _{x \rightarrow \infty} f(x)=\infty$ THEN $\lim _{x \rightarrow \infty} \sin (f(x))$ does not exist.

I know that this statement is false.

I will give a counterexample for this as just take $f(x) = \lfloor x \rfloor \pi$.
How to prove that using formal definition of limit that this example violate above statement.