Question

The set of all values of 'a' for which $\underset{x\to a}{lim}\left[x\right]$ does not exist is ($\left[x\right]$ denotes greatest integer function)

• A. a is any integer
• B. a is a positive rational number
• C. a is a negative rational integer
• D. a is complex number

Answer 1

The correct option is A a is any integer

Let us assume that $a=k$, where $k$ is an integer.
When, $x\to a^{-}$
$k-1<x<k$
Or, $\underset{x\to a^{-}}{lim}\left[x\right]=k-1$
Similarly,When, $x\to a^{+}$
$k<x<k+1$
Thus, $\underset{x\to a^{+}}{lim}\left[x\right]=k$
Since LHL and RHL are not equal, the limit does not exist, when $a$ is an integer.