Q. $\lim_{x \rightarrow 0}( - 1)^{\lbrack x\rbrack}$, where [.] denotes the greatest integer function is equal to
A) 0
B) 1
C) -1
D) Does not exist
Explanation
(d)
LHL$= \lim_{x \rightarrow 0^{-}}( - 1)^{\lbrack x\rbrack}$
$= \lim_{h \rightarrow 0}( - 1)^{\lbrack 0 - h\rbrack} = ( - 1)^{- 1} = - 1$
RHL$= \lim_{x \rightarrow 0^{+}}( - 1)^{\lbrack x\rbrack} = \lim_{h \rightarrow 0}( - 1)^{\lbrack 0 + h\rbrack}$
$= ( - 1)^{0} = 1$
$\because$ LHL$\neq$RHL
$\therefore$ Limit does not exist.
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.