`lim_(xrarr0) x^8[(1)/(x^3)]`, where `[.]`,denotes the greatest integer function is

Options

A. a non-zero positive real number
B. a negative real number
C. 0
D. non-existent

Answer

C

Explanation

For any `x in R`, we have
`x-1lt [x]le x`
` rArr (1)/(x^3)-1lt [(1)/(x^3)]le(1)/(x^3)"for all " 0 ne in R`
` rArr x^8((1)/(x^3-1))lt x^8[(1)/(x^3)]le x^5 " for all " x (ne 0) in R`
`rArr x^5-x^8lt x^8[(8)/(x^3)]le x^5" for all " 0 ne x in R`
But , `lim_(xto0) x^5-x^8=0and lim_(xto0) x^5=0`.
So, by Sandwich theorem, we have
`lim_(xto0) x^8[(1)/(x^3)]=0`.