Question

Find the value of$\underset{x\to 2}{lim}\left[x\right]$, where [x] represents greatest integer less than or equal to x.

• A. 2
• B. 3
• C. 1
• D. None of these

Answer 1

The correct option is D

None of these

lets look at the greatest integer before we solve this. The graph of [x] looks like.

We can see that the garph breaks at every integer points.

We want to find $\underset{x\to 2}{lim}\left[x\right]$

We will find left hand limit $(\underset{x\to 2}{lim}-f(x\left)\right)$ and Right hand limit $(\underset{x\to 2}{lim}+f(x\left)\right)$ separately, because the

garph breaks at the point where we want to find the limit.

$(\underset{x\to 2}{lim}-f(x\left)\right)=\underset{h\to 0}{lim}f(2-h)=1$
& $(\underset{x\to 2}{lim}+f(x\left)\right)=\underset{h\to 0}{lim}f(2+h)=2$
Since, Both limits are not equal the limit doesn't exist.
Note - In greatest integer function limit doesn't exist at any integer point.