F(x) is the floor function.
#limit
robust roost
steep cometBOT
carmine venture
Doesn’t it approach 0?
Because no matter what you do, that sin(x) will give you 0
echo hamlet
You cant just say that because it is of the form $0\cdot -\infty$
rugged caveBOT
Ryanstaal2006
flat hearth
yea 0 times inf is undetermined
robust roost
Because no matter what you do, that sin(x) will give you 0
it is undefined
by graphing the function on Desmos , i can see it is -1 , but i have no idea how to get to this result
flat hearth
technically floor(1/x) ≈ 1/x
so -sin(x) * floor(1/x) ≈ -sin(x)/x = 1 if x -> 0
Maybe squeeze theorem works
robust roost
technically floor(1/x) ≈ 1/x
so -sin(x) * floor(1/x) ≈ -sin(x)/x = 1 if x -> 0
whart do you mean by that
floor(x) ≈ 1 / x ????
flat hearth
floor(x) ≈ 1 / x ????
sorry i meant that the floor(1/x) behaves similiarly to 1/x
robust roost
in term of the limit ?
orchid terrace
consider x is a natural number and resolve it
robust roost
how would that work?
orchid terrace
or consider x is a natural number n + a real number between 0 and 1
robust roost
i don't understand
orchid terrace
ok i got it
$\lim_{x\rightarrow 0} \sin(x)\left[ x - \lfloor \frac{1}{x} \rfloor \right]=\lim_{x\rightarrow 0}x\sin(x) - \lim_{x\rightarrow 0} \sin(x)\lfloor \frac{1}{x} \rfloor=-\lim_{x\rightarrow \infty} \sin(\frac{1}{x})\left( x - { x } \right)$
rugged caveBOT
tio zuca
robust roost
there still is a problem to calculate the second limit
orchid terrace
and then evaluated the xsinx at 0
the final limit i substituted x to 1/x, so the limit now is infinity
and the {x} is the fractional part of x, like if x = 3.14 then {x}=0.14
distribute again and you will have lim {x} sin(1/x) - xsin(1/x)
the limit xsin(1/x) at x->infinity is the same as 1/x*sin(x) at x-> 0
sin(x)/x -> 1 as x-> 0
so is {x}sin(1/x) - 1
{x} is always between 0 and 1
orchid terrace
if x approaches infinity being natural numbers like 1, 2, 3, 4, ... up to infinity (Like a sequence) {x} is 0 since it don't have a fractional part
robust roost
that's a completly different approach from mine
orchid terrace
if x approaches infinity being just fractional numbers like 0.75, 5.32456, etc etc, {x} isnt 0, so you just evaluate sin(1/x) at x-> infty which is 0
so the answer is -1
robust roost
thank you for the explanation kind sir
orchid terrace
i'm sorry if i did some grammar errors, i'm brazilian and i dont speak english very well
u r welcome
robust roost
no problem , you speak well
orchid terrace
thanks
robust roost
.solved