Old-question-related-to-greatest-int-function-lim-x-0-1-x-1-1-1-lim-x-0-1-x-

Question Number 2624 by prakash jain last updated on 23/Nov/15

Old question related to greatest int function .

\lim_{x \to 0} ⌊ 1 + x ⌋ = 1

⌊ 1 ⌋ = 1

\lim_{x \to 0} ⌊ 1 - x ⌋ = ?

Answered by Filup last updated on 24/Nov/15

if ⌊ a \pm b ⌋ = ⌊ a ⌋ \pm ⌊ b ⌋ : for (a, b) \in Z

\lim_{x \to 0} ⌊ 1 + x] = \lim_{x \to 0} ⌊ 1 ⌋ + \lim_{x \to 0} ⌊ x ⌋

= 1 + \lim_{x \to 0} ⌊ x ⌋ = 1 + ⌊ 0 ⌋ = 1

Similarly

\lim_{x \to 0} ⌊ 1 - x ⌋ = \lim_{x \to 0} ⌊ 1 ⌋ - \lim_{x \to 0} ⌊ x ⌋

= 1 - ⌊ 0 ⌋ = 1

Commented by Yozzi last updated on 24/Nov/15

L e t a = 1.99, b = 1.01

\Rightarrow ⌊ 1.99 + 1.01 ⌋ = ⌊ 3 ⌋ = 3 \neq ⌊ 1.99 ⌋ + ⌊ 1.01 ⌋ = 1 + 1 = 2

Commented by Filup last updated on 24/Nov/15

T hanks for the proof!

B ack to the drawing board!

Commented by Filup last updated on 24/Nov/15

i f ⌊ a \pm b ⌋ = ⌊ a ⌋ \pm ⌊ b ⌋ for (a, b) \in Z

t h e n m y a n s w e r i s t r u e

I will edit my post!

if a, b \in Z, ⌊ a ⌋ = a, ⌊ b ⌋ = b

Answered by Yozzi last updated on 24/Nov/15

L e t y = ⌊ 1 - x ⌋ . \lim_{x \to 0^{+}} y = \lim_{x \to 0^{+}} ⌊ 1 - x ⌋

F o r 0 ⩽ x < 1, 0 < 1 - x ⩽ 1 \Rightarrow 1 - x i s a

f r a c t i o n o r 1 \Rightarrow ⌊ 1 - x ⌋ = 0 o r ⌊ 1 - x ⌋ = 1

\Rightarrow \lim_{x \to 0^{+}} ⌊ 1 - x ⌋ i s u n d e f i n e d (c a n n o t b e s u r e t o c h o o s e 1 o r 0)

F o r - 1 < x ⩽ 0, 1 ⩽ 1 - x < 2 \Rightarrow ⌊ 1 - x ⌋ = 1 .

∴ \lim_{x \to 0^{-}} ⌊ 1 - x ⌋ = 1 . \lim_{x \to 0^{-}} y \neq \lim_{x \to 0^{+}} y s i n c e \lim_{x \to 0^{+}} y d o e s n o t e x i s t .

\Rightarrow \lim_{x \to 0} y d o e s n o t e x i s t . G r a p h i c a l l y,

t h e r e i s a j u m p d i s c o n t i n u i t y i n

y = ⌊ 1 - x ⌋ a t x = 0 .

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