The discussion centers around evaluating the limit of \((\sin(2x))^3/(\sin(3x))^3\) as \(x\) approaches 0, a topic within calculus focusing on limits and trigonometric functions.
Some participants have provided hints and guidance on how to approach the limit using the known limit of \(\sin(x)/x\). There is ongoing exploration of how to apply this concept to the original limit problem, with no explicit consensus reached yet.
Participants have expressed a desire for a worked-out answer, but the forum guidelines discourage providing complete solutions. The discussion reflects uncertainty about how to manipulate the sine functions in the limit context.
Nitrate said:Homework Statement
evaluate the limit as x approaches 0 of (sin(2x))^3/(sin(3x))^3
Homework Equations
The Attempt at a Solution
the answer in the textbook is 8/27
i figured they got that by raising 2 to the 3rd and 3 to the 3rd
but I'm not entirely sure what happens to the sin's
id really appreciate a worked out answer (that doesn't use l'hopital's rule)
LCKurtz said:We don't do worked out answers. But here's a hint: Presumably you know something about the limit of sin(x)/x as x → 0. Make use of that by multiplying and dividing by certain powers of x.
Nitrate said:well
lim
x->0 sinx/x = 1
the problem is that i don't know how to apply it to this question
that limit would be 1 as wellLCKurtz said:What would be the limit of [itex]\frac {\sin(2x)}{2x}[/itex] as x → 0?
Nitrate said:that limit would be 1 as well
but I'm still not sure how it'd help