Question

$\underset{x\to 1}{\lim }\left[x-1\right]$, where [.] is the greatest integer function, is equal to

(a) 1 (b) 2 (c) 0 (d) does not exist

Answer 1

We have,

$\underset{x\to 1^{-}}{\lim }\left[x-1\right]=\underset{h\to 0}{\lim }\left[1-h-1\right]=\underset{h\to 0}{\lim }\left[-h\right]=-1$ (k − 1 < k − h < k ⇒ [k − h] = k − 1, k ∈ Z)

Also,

$\underset{x\to 1^{+}}{\lim }\left[x-1\right]=\underset{h\to 0}{\lim }\left[1+h-1\right]=\underset{h\to 0}{\lim }\left[h\right]=0$

$\therefore \underset{x\to 1^{-}}{\lim }\left[x-1\right]\ne \underset{x\to 1^{+}}{\lim }\left[x-1\right]$

Thus, $\underset{x\to 1}{\lim }\left[x-1\right]$ does not exist.

Hence, the correct answer is option (d).