Question

If $f(x)=\begin{cases}x+\frac{1}{2}, & x<0 \\ 2 x+\frac{3}{4}, & x \geq 0\end{cases}$,
then $\left[\displaystyle\lim _{x \rightarrow 0} f(x)\right]=($ where [.] denotes the greatest integer function)

• A. $\frac{1}{2}$
• B. $\frac{3}{4}$
• C. does not exist
• D. none of these

Answer 1

Answer: (C)

$\displaystyle\lim _{x \rightarrow 0^{-}} f(x)=\displaystyle\lim _{x \rightarrow 0^{-}}\left(x+\frac{1}{2}\right)=\frac{1}{2}$
$\displaystyle\lim _{x \rightarrow 0^{+}} f(x)=\displaystyle\lim _{x \rightarrow 0^{+}}\left(2 x+\frac{3}{4}\right)=\frac{3}{4}$
$\therefore \displaystyle\lim _{x \rightarrow 0} f(x)$ does not exist.