Question

How do you find the limit of `(sinx)/(3x)` as x approaches `oo`?

Answer 1

0

Explanation:

you can see sinx as a wave picture whose value is always between 1 and -1
graph{sinx [-10, 10, -5, 5]}

and then see 3x

when x becomes larger and larger in denominator

we dont need to care the value of sinx

it will become 0

Answer 2

`lim_(xto +-oo) sin(x)/(3x)=0`

Explanation:

`color(blue)("Consider "sin(x) )`

This function can assume all values between and including -1 and +1. It will repeat this cycle every `2pi` radians as appropriate for the value of `x`. So `sin(x) in (-1 , +1)`
Where `(-1,+1)` is the range of all value between and including -1 to +1.

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Consider "3x)`

As `x` becomes increasing larger in the positive or negative direction, `1/(3x)` becomes increasingly smaller.

So for `x<0" "sin(x)/(3x)" tends to 0 but on the negative side of 0"`

So for `x>0" "sin(x)/(3x)" tends to 0 but on the positive side of 0"`

Thus `lim_(xto +-oo) sin(x)/(3x)=sin(x)/oo = 0`

As you move further and further away from the origin the amplitude lessens and approaches `y=0`