lim x arrow 3([x-3]+[3-x]-x), where [⋅] denotes the greatest integer function, is equal to

Question

lim x arrow 3([x-3]+[3-x]-x), where [⋅] denotes the greatest integer function, is equal to (A) 4 (B) -4 (C) 0 (D) does not exist

Tardigrade Admin · Accepted Answer

displaystyle lim x arrow 3([x-3]+[3-x]-x) = displaystyle lim x arrow 3([x]-3+3+[-x]-x) = displaystyle lim x arrow 3([x]+[-x]-x)=-1-3=-4 (∵[x]+[-x]=-1, when x ∉ I)

Q. $lim_{x \to 3} ([x - 3] + [3 - x] - x)$ , where $[\cdot]$ denotes the greatest integer function, is equal to

4

-4

0

does not exist

$x \to 3 lim ([x - 3] + [3 - x] - x)$
$= x \to 3 lim ([x] - 3 + 3 + [- x] - x)$
$= x \to 3 lim ([x] + [- x] - x) = - 1 - 3 = - 4$
$(∵ [x] + [- x] = - 1$ , when $x \in / I)$

Q. limx→3​([x−3]+[3−x]−x), where [⋅] denotes the greatest integer function, is equal to

4

-4

0

does not exist

x→3lim​([x−3]+[3−x]−x) =x→3lim​([x]−3+3+[−x]−x) =x→3lim​([x]+[−x]−x)=−1−3=−4 (∵[x]+[−x]=−1, when x∈/I)

Q. $lim_{x \to 3} ([x - 3] + [3 - x] - x)$ , where $[\cdot]$ denotes the greatest integer function, is equal to

$x \to 3 lim ([x - 3] + [3 - x] - x)$
$= x \to 3 lim ([x] - 3 + 3 + [- x] - x)$
$= x \to 3 lim ([x] + [- x] - x) = - 1 - 3 = - 4$
$(∵ [x] + [- x] = - 1$ , when $x \in / I)$