Question

If $l\left(x\right)$ is the least integer not less than $x$ and $g\left(x\right)$ is the greatest integer not greater than $x$, then $\underset{x\to e+\mathrm{\pi }}{\lim }\left\{l\left(x\right)+g\left(x\right)\right\}=?$

• A. $9$
• B. $13$
• C. $1$
• D. None of these

Answer 1

The correct option is D

None of these

Explanation for the correct option:

It is given that,

$l\left(x\right)\to$ least integer $\ge x$, and

$g\left(x\right)\to$ greatest integer $\le x$

Now it is given that for the limit $x$ tends to $e+\mathrm{\pi }$

So, we know that, $e=2.71,\mathrm{\pi }=3.14$

Thus, $x=e+\mathrm{\pi }=5.85$

For, $l\left(x\right)=l(5.85)$, the least integer will be $6$

and for, $g\left(x\right)=g(5.85)$, the greatest integer will be $5$.

Thus,

$$\begin{gathered} \Rightarrow \left\{l\left(x\right)+g\left(x\right)\right\}=5+6 \\ \Rightarrow =11 \end{gathered}$$

Therefore, the correct option is (D), “None of these”.