if f(x)= { ((((sin (1+[x]))/([x])) for [x]≠0)),((0 for [x]=0)) :} where [x] represents an integer x greatest ≤ x Find lim


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Question Number 201657 by LimPorly last updated on 10/Dec/23
$if f(x)= { ((((sin (1+[x]))/([x])) for [x]≠0)),((0 for [x]=0)) :} where [x] represents an integer x greatest ≤ x Find lim_(x→0^− ) f(x).$
$i f f (x) = {\begin{cases} \frac{\sin (1 + [x])}{[x]} f o r [x] \neq 0 \\ 0 f o r [x] = 0 \end{cases}$ $w h e r e [x] r e p r e s e n t s a n i n t e g e r x g r e a t e s t ⩽ x$ $F i n d \lim_{x \to 0^{-}} f (x) .$
	Answered by aleks041103 last updated on 10/Dec/23
	$If I understand correctly: [x]=max{n:n∈Z ∧ x≥n} i.e. [2]=2; [3.5]=3; [−1.5]=−2 and so on. Then lim_(x→0^− ) [x]=lim_(ε→0^+ ) [0−ε]=lim_(ε→0^+ ) [−ε]=−1 ⇒lim_(x→0^− ) f(x)=( { ((((sin (1+[x]))/([x])) for [x]≠0)),((0 for [x]=0)) :})_([x]=−1) = =((sin(1+(−1)))/((−1)))=((sin(0))/(−1))=0$
	$I f I u n d e r s t a n d c o r r e c t l y :$ $[x] = m a x {n : n \in Z \land x ⩾ n}$ $i . e .$ $[2] = 2; [3.5] = 3; [- 1.5] = - 2 a n d s o o n .$ $T h e n$ $\underset{x \to 0^{-}}{l i m} [x] = \underset{ε \to 0^{+}}{l i m} [0 - ε] = \underset{ε \to 0^{+}}{l i m} [- ε] = - 1$ $\Rightarrow \underset{x \to 0^{-}}{l i m} f (x) = {({\begin{cases} \frac{\sin (1 + [x])}{[x]} f o r [x] \neq 0 \\ 0 f o r [x] = 0 \end{cases})}_{[x] = - 1} =$ $= \frac{s i n (1 + (- 1))}{(- 1)} = \frac{s i n (0)}{- 1} = 0$
	Commented by LimPorly last updated on 10/Dec/23

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