Question

limx->0 (x/abs(x)) . sinx

Answer 1

seems unlikely

Answer 2

\[\frac{x}{|x|}\] is 1 if \(x>0\) and \(-1\) if \(x<0\)

Answer 3

Why is that? Wouldn't Sin(0) = 0 ?

Answer 4

oooh my bad (as we say) i didn't see the \(\sin(x)\) there!!

Answer 5

it is zero for sure, but there is something to say, because your expression is not defined at \(x=0\)

Answer 6

you have to take a right and a left sided limit
\[\lim_{x\to 0^+}\sin(x)=0\] and
\[\lim_{x\to 0^-}\sin(x)=0\]

Answer 7

you have to take them both because your expression is \(\sin(x)\) if \(x>0\) and it is \(-\sin(x)\) if \(x<0\)

Answer 8

so ... The answer should be like this? Right?
limx→0+sin(x)=0
and
limx→0−sin(x)=0

@@satellite73

Answer 9

we can use this relation \[|\sin(x)|\leq x\] and then multiply by \[|\frac{x}{|x|}|\]
we will obtain \[|\frac{x}{|x|}\sin(x)|\leq x\] and then the limit as \[x\to 0\] the limit is zero

Answer 10

@Mega96 yes