limx→1 ((1−x)+[x−1]+|1−x|) where ∣[x] denotesthe greatest integer less than or equal to x

$lim_{x \to 1} ((1 - x) + [x - 1] + | 1 - x |)$ where $∣ [x]$ denotes
the greatest integer less than or equal to x

For $0 < x < 1, - 1 < x - 1 < 0$ , so

$F (x) = (1 - x) + (- 1) + (1 - x) = 1 - 2 x$

$\Rightarrow lim_{x \to 1 -} f (x) = 1 - 2 . (1) = - 1$ .

For $1 < x < 2, 0 < x - 1 < 1$ , so

$f (x) = (1 - x) + 0 + (x - 1) = 0$

$\Rightarrow lim_{x \to 1 +} f (x) = 0$ Thus $lim_{x \to | |} f (x)$ does not exist.

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