What is the limit as x approaches 0 of sin(3x)/sin(4x)?

1 Answer
Dec 18, 2014

The answer is 3/4.

You both sin(3*0) = 0 and sin(4*0)=0, so you can use l'hopitals rule. This is:
lim_(x->0) sin(3x)/sin(4x) = lim_(x->0) ([sin(3x)]')/([ sin(4x)]') = lim_(x->0) (3cos(3x))/(4cos(4x)) =(3cos(0))/(4cos(0)) = 3/4

The first step is taking the derivative of both the nominator and the denominator; the last step is just filling in zero, you're allowed to do this because cos(0) = 1, so you don't risk dividing by zero.