Answer
(a) $400$
(b) $399$
(c) Does not exist.
Work Step by Step
(a) For the Greatest Integer function, within $400\lt x \lt 400+\delta, 0\lt\delta\lt1$, we have $\lfloor x\rfloor=400$. Thus for any small number $\epsilon\gt0$, we can choose $\delta=0.5$ so that for all $x$ in the interval of $400\lt x \lt 400+0.5$, we have $|\lfloor x\rfloor-400|=0\lt\epsilon$, which proves that $\lim_{x\to400^+}\lfloor x\rfloor=400$
(b) Similarly, within $400-\delta\lt x\lt 400, 0\lt\delta\lt1$, we have $\lfloor x\rfloor=399$. Thus for any small number $\epsilon\gt0$, we can choose $\delta=0.5$ so that for all $x$ in the interval of $400-0.5\lt x\lt 400$, we have $|\lfloor x\rfloor-399|=0\lt\epsilon$, which proves that $\lim_{x\to400^-}\lfloor x\rfloor=399$
(c) Because $\lim_{x\to400^-}\lfloor x\rfloor\ne\lim_{x\to400^+}\lfloor x\rfloor$, $\lim_{x\to400}\lfloor x\rfloor$ does not exist.
