To evaluate the limit as x approaches 0 of (tan^3(2x)/sin^2(3x)), we can use trigonometric identities to simplify the expression.
First, we know that tan(x) = sin(x)/cos(x). So, tan^3(x) = (sin(x)/cos(x))^3 = sin^3(x)/cos^3(x).
Similarly, sin(3x) = 3sin(x) - 4sin^3(x), so sin^2(3x) = (3sin(x) - 4sin^3(x))^2.
Now let's substitute these expressions back into the original limit:
lim (x→0) (tan^3(2x)/sin^2(3x))= lim (x→0) (sin^3(2x)/cos^3(2x)) / ((3sin(x) - 4sin^3(x))^2)
Since we are taking the limit as x approaches 0, we can use the fact that sin(x) ~ x and cos(x) ~ 1 for small x.
lim (x→0) (sin^3(2x)/cos^3(2x)) / ((3sin(x) - 4sin^3(x))^2)= lim (x→0) ((2x)^3/(1)^3) / ((3x - 4x^3)^2)= lim (x→0) (8x^3) / (9x^2 - 24x^4 + 16x^6)= 0
Therefore, lim (x→0) (tan^3(2x)/sin^2(3x)) = 0.
To evaluate the limit as x approaches 0 of (tan^3(2x)/sin^2(3x)), we can use trigonometric identities to simplify the expression.
First, we know that tan(x) = sin(x)/cos(x). So, tan^3(x) = (sin(x)/cos(x))^3 = sin^3(x)/cos^3(x).
Similarly, sin(3x) = 3sin(x) - 4sin^3(x), so sin^2(3x) = (3sin(x) - 4sin^3(x))^2.
Now let's substitute these expressions back into the original limit:
lim (x→0) (tan^3(2x)/sin^2(3x))
= lim (x→0) (sin^3(2x)/cos^3(2x)) / ((3sin(x) - 4sin^3(x))^2)
Since we are taking the limit as x approaches 0, we can use the fact that sin(x) ~ x and cos(x) ~ 1 for small x.
lim (x→0) (sin^3(2x)/cos^3(2x)) / ((3sin(x) - 4sin^3(x))^2)
= lim (x→0) ((2x)^3/(1)^3) / ((3x - 4x^3)^2)
= lim (x→0) (8x^3) / (9x^2 - 24x^4 + 16x^6)
= 0
Therefore, lim (x→0) (tan^3(2x)/sin^2(3x)) = 0.