Limit of Trig Function

[thunc14]

So I'm a bit stuck on this problem. As you can see from my work, I used some trig identities to manipulate tan(2x) to sin(2x)/cos(2x) in the numerator. I then rewrote sin(2x) as 2sin(x)cos(x), which I'm sure is necessary or not to solve the problem. I still have a 1/x when I separate the terms. I'm stuck at this step and I'm not quite sure where to go. Any suggestions? Thanks in advance.

 

[pka]

\(\mathop {\lim }\limits_{x \to 0} \dfrac{{\tan (2x)}}{{3x\cos (4x)}} = \mathop {\lim }\limits_{x \to 0} \dfrac{{\sin (2x)}}{{3x}} \cdot \mathop {\lim }\limits_{x \to 0} \dfrac{1}{{\cos (2x) \cdot \cos (4x)}}\)

 

[thunc14]

\(\mathop {\lim }\limits_{x \to 0} \dfrac{{\tan (2x)}}{{3x\cos (4x)}} = \mathop {\lim }\limits_{x \to 0} \dfrac{{\sin (2x)}}{{3x}} \cdot \mathop {\lim }\limits_{x \to 0} \dfrac{1}{{\cos (2x) \cdot \cos (4x)}}\)

I got that, but I'm confused as to what to do with the x that's in the denominator (sin(2x)/(3x)). I'm sure I'm just overlooking something simple but I'm stuck at that step.

 

[Deleted member 4993]

I got that, but I'm confused as to what to do with the x that's in the denominator (sin(2x)/(3x)). I'm sure I'm just overlooking something simple but I'm stuck at that step.

Hint:

sin(2x)/(3x) = 2/3 * sin(2x)/(2x)

 

[thunc14]

Hint:

sin(2x)/(3x) = 2/3 * sin(2x)/(2x)

So that's just a little manipulation that would simplify down to the original expression? I'm still lost looking at it because no matter how I write it out I seem to always have an x in the denominator left over.

 

[Deleted member 4993]

So that's just a little manipulation that would simplify down to the original expression? I'm still lost looking at it because no matter how I write it out I seem to always have an x in the denominator left over.

Recall:

[math]\lim_{\theta \to 0}\left [\frac{\sin (\theta )}{\theta}\right] = 1[/math]

 

[thunc14]

Ah, there's the missing piece of information. I'll try to look up other properties like that to learn. I didn't know that that was equal to 1. Thank you very much for the help. I see that (1-cos(x))/x is another common one, anything else worth looking into?

 

[Deleted member 4993]

Ah, there's the missing piece of information. I'll try to look up other properties like that to learn. I didn't know that that was equal to 1. Thank you very much for the help. I see that (1-cos(x))/x is another common one, anything else worth looking into?

tan(x)/x