Need verification to solution for trig function limit question

[kylera]

As I do not have the solution book and classes are over for the summer, I don't have any other way of verifying whether this is right or not.

Homework Statement

lim(x->0) (sin 4x) / (sin 6x)

Homework Equations

The Attempt at a Solution

If I may temporarily remove the lim sign...
$$\frac{sin 4x}{sin 6x} = \frac{sin4x}{4x} \times \frac{4x}{sin6x}$$

$$= \frac{sin4x}{4x} \times \frac{6x}{sin6x} \times \frac{4x}{6x}$$

By Limit Laws, take the limit of each fraction to get
$$1 \times 1 \times \frac{4}{6}$$

4. Final answer
2/3

 

[Defennder]

You could also use L'Hopital rule to get the same result.