Question

53. (a) if the symbol
`\lbrack\rbrack` denotes the greatest integer function defined in example 10 , evaluate

(i)
`\lim _(x\rightarrow-2^+)\lbrack x\rbrack`
(ii)
`\lim _(x\rightarrow-2)\lbrack x\rbrack`
(iii)
`\lim _(x\rightarrow-2.4)\lbrack x\rbrack`

Answer 1

If x is between two consecutive integers such that n ≤ x < n + 1, then the greatest integer function [x] maps x to the largest integer smaller than x so that [x] = n.

(i) If x is approaching -2 from above, that means x > -2. As x gets closer to -2, we essentially have -2 < x < -1, so that [x] will approach

`\displaystyle \lim_(x\to-2^+) [x] = \boxed{-2}`

(ii) However, if x is approaching -2 from below, then x < -2, so that [x] = -3. In other words

`\displaystyle \lim_(x\to-2^-) [x] = -3 \\eq -2`

Because the one-sided limits do not match, the two-sided limit

`\displaystyle \lim_(x\to-2) [x] ~~\boxed{\text{does not exist}}`

(iii) -2.4 lies between -3 and -2, so

`\displaystyle \lim_(x\to-2.4) [x] = \boxed{-3}`