The limit of (sin(2x))^3/(sin(3x))^3 as x approaches 0 evaluates to 8/27. This result is derived from the known limit of sin(x)/x as x approaches 0, which equals 1. By applying this concept, one can manipulate the expression by multiplying and dividing by appropriate powers of x to simplify the limit calculation without using L'Hôpital's rule.
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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 \frac {\sin(2x)}{2x} as x → 0?
Nitrate said:that limit would be 1 as well
but I'm still not sure how it'd help