Question

lim_x→1[x−1], Where [.] is the greatest integer function, is equal to

Answer 1

We have, lim_x→1⁻ [x−1] = lim_h→0 [1−h−1] = lim_h→0[−h] = −1

(k − 1 < k − h < k ⇒ [k − h] = k−1k ϵ Z)

Also,

lim_x→1+ [x−1] = lim_h→0[1+h−1] = lim_h→0[h] = 0

∴ lim_x→1−[x−1] ≠ lim_x→1+[x−1]