How do you find the limit of (sinx)/(3x)sinx3x as x approaches oo?

2 Answers
Jun 16, 2016

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

Jun 16, 2016

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

Tony BTony B

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