Question

53. (a) if the symbol \rbrack" align="absmiddle" class="latex-formula"> 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

$\displaystyle \lim_{x\to-2^-} [x] = -3 \neq -2$

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

$\displaystyle \lim_{x\to-2.4} [x] = \boxed{-3}$