Finding the Limit of Trig Functions: 2x/sin3x, x->0

[domyy]

Homework Statement

lim 2x/(sin3x)
x-> 0

Homework Equations

lim sinx/x = 1
x->0

The Attempt at a Solution

is it correct to say the following:

lim 2/3 (sinx/x)
x-> 0

lim 2/3 (1)
x-> 0

Answer: lim 2x/sin3x = 2/3
x-> 0

Because it's on the book:
cos 2x(2)/3 = 2/3
x->0

So I want to know if I solve it the way I did before, would it be correct?

 

[ehild]

Homework Statement

lim 2x/(sin3x)
x-> 0

Homework Equations

lim sinx/x = 1
x->0

The Attempt at a Solution

is it correct to say the following:

lim 2/3 (sinx/x)
x-> 0

lim 2/3 (1)
x-> 0

Answer: lim 2x/sin3x = 2/3
x-> 0

Because it's on the book:
cos 2x(2)/3 = 2/3
x->0

So I want to know if I solve it the way I did before, would it be correct?

lim 2/3 sin(x)/x is zero at x=0, but 2x/sin(3x) is not the same as (2/3)sin(x)/x.

Rewrite 2x/sin(3x) as 2/3 (3x)/sin(3x).

ehild

 

[domyy]

Thanks! Got it! =)