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
The limit of (sin(2x))^3/(sin(3x))^3 as x approaches 0 is equal to 8/27.
To solve this limit, we can use the L'Hopital's rule, which states that the limit of f(x)/g(x) as x approaches a is equal to the limit of f'(x)/g'(x) as x approaches a, where f'(x) and g'(x) are the derivatives of f(x) and g(x) respectively.
The limit of (sin(2x))^3/(sin(3x))^3 as x approaches 0 is significant because it is the value that the function approaches as x gets closer and closer to 0. It helps us understand the behavior of the function near the point x = 0.
The limit of (sin(2x))^3/(sin(3x))^3 as x approaches 0 is not defined because the function is undefined at x = 0. This is because the denominator, (sin(3x))^3, becomes 0 at x = 0, which results in a division by 0 error.
The value of the limit of (sin(2x))^3/(sin(3x))^3 as x approaches 0 remains the same regardless of whether we use radians or degrees. This is because the trigonometric functions, sin(2x) and sin(3x), are defined in terms of radians and degrees, and their values are equal at x = 0 in both units.