lim of sin 4x / sin 6x as x approaches 0. I'm pretty sure the answer is 2/3, but I need a method of doing it. Any help is appreciated.
Well, put it this way:
$\displaystyle \frac{\sin(4x)}{\sin(6x)} = \frac{\sin(4x)}{4x} \times \frac{6x}{\sin(6x)} \times \frac{4x}{6x} $
The xs cancel, and 4/6 simplifies to 2/3
$\displaystyle \frac{\sin(4x)}{\sin(6x)} = \frac{\sin(4x)}{4x} \times \frac{6x}{\sin(6x)} \times \frac{2}{3} $
Now, do you know that $\displaystyle \displaystyle \lim_{x \to 0} \frac{\sin(nx)}{nx} = 1 $. If so, then the limit we have is trivial now .