Question

Compute $\underset{x\to 2+}{lim}\left(\right[x]+x)$ and $\underset{x\to 2-}{lim}\left(\right[x]+x)$.

Answer 1

$\underset{x\to 2^{+}}{lim}\left(\right[x]+x)$

for $x\to 2^{+};\left[x\right]=2$

$\therefore \underset{x\to 2^{+}}{lim}\left(\right[x]+x)=\underset{x\to 2^{+}}{lim}(2+x)=4$

$\underset{x\to 2^{-}}{lim}\left(\right[x]+x)$

for $x\to 2^{-};\left[x\right]=1$

$\therefore \underset{x\to 2^{-}}{lim}\left(\right[x]+x)=1+2=3$