Question Number 105036 by bobhans last updated on 25/Jul/20

lim_(x→0) x.[ (1/x) ] ?  note [ ] = greatest integer function

Answered by john santu last updated on 25/Jul/20

the limit as x→0^+  can be  transformed into lim_(p→∞) ((⌊ p ⌋)/p)   set { p } = p − ⌊ p ⌋   we have ((⌊ p ⌋)/p) = 1− (({p})/p) , since  0≤ {p} < 1 . so lim_(p→∞) (1−(({p})/p))=  1−0 = 1 .similarly for  the limit as x→0^−    (JS ♠⧫)

Answered by mathmax by abdo last updated on 25/Jul/20

we have [(1/x)] ≤(1/x)<[(1/x)] +1 ⇒ for x>0 ⇒x[(1/x)]≤1<x[(1/x)]+x ⇒   { ((x[(1/x)]≤1  ⇒   1−x <x[(1/x)]≤1  we passe to limit  (x→o^+ ) ⇒)),((1−x<x[(1/x)])) :}  lim_(x→0^+ )    x[(1/x)] =1