Question

Let $f\left(3\right)=4$ and $f^{\prime }\left(3\right)=5.$ Then , $\underset{x\to 3}{lim}\left[f\left(x\right)\right]$ where $[\cdot ]$ denotes the greatest integer function, is

• A. $3$
• B. $4$
• C. $5$
• D. does not exist

Answer 1

Answer: (D)

does not exist

$f^{\prime }\left(3\right)=5$
$\Rightarrow \underset{h\to 0}{lim}\frac{f(3+h)-f\left(3\right)}{h}=5$
If $h>0$, then $f\left(3^{+}\right)>4$, $⟹\left[f\right(3+h\left)\right]=4$
If $h<0$, then $f\left(3^{-}\right)<4$, $⟹\left[f\right(3-h\left)\right]=3$
Hence limit of $\left[f\right(x\left)\right]$ does not exit at $x=3$