Calculus, find the limit

Question

Find the limit of sin3x/(5x^3 - 4x) as x goes to zero.

Answer 1

In case you don't know L'Hopital's Rule, we can do the following:

sin(3x) / (5x³ - 4x) = sin(3x) / [x(5x²-4)] = 3[sin(3x)/(3x)]{1/(5x²-4)]

So, taking the limit as x approaches 0, we have 3(1)(1/(-4)) = -3/4

Recall that the limit of sinθ/θ as θ approaches 0 is equal to 1.

Answer 2

If we plug in zero for the expression, we end up with 0/0. That means we can apply L'Hopital's Rule. Taking the derivative of the numerator and denominator gives us (3cos3x) / (15x² - 4). If we plug in 0 here, we get -3/4. So the limit is -3/4.