Tricky Limit as x approaches zero

[cal.queen92]

Homework Statement

find the limit as x approaches zero of: (sin(4x))/(tan(3x))

Homework Equations

a law that states : limit as x approaches zero of (sin(theta))/theta = 1

The Attempt at a Solution

I first divide the equation into:

(sin(4x))/1 * (cos(3x))/(sin(3x))

And then use the law mentioned above to multiply and dive by 4x:

(sin(4x))/(4x) * 4x(cos(3x))/(sin(3x))

Then, I divide the equation into:

4x*(cos(3x))/1 * 1/(sin(3x))

Then, I use the law again on cos(3x) this time:

4x* (cos(3x))/3x * 3x/(sin(3x))

And end up with:

4x * 3x/(sin(3x))

That's where I get stuck!

Am I using the right method? If so, where do i go next?

Thank you!

 

[iainfs]

Try expanding numerator and denominator as Maclaurin series.

 

[cal.queen92]

Okay, but first, which numerator and denominator? The original equation, or my final one?

Also, what is a Maclaurin series? I've never heard of it before...

 

[stevenb]

Also, what is a Maclaurin series? I've never heard of it before...

Specific case of Taylor series with expansion around zero.

For example,

sin(x)=x-x^3/6+x^5/120 ...

and

tan(x)=x+x^3/3+2x^5/15 + ...

http://orion.math.iastate.edu/vika/cal3_files/Lec27.pdf

 

[snipez90]

If you're not going to use LaTeX, you should at least make it clear which terms are in the numerator or denominator, especially since you seemed to have made an error in leap to the last line.

You're somewhat fine up to:

$$\lim_{x \rightarrow 0}\frac{4x \cos 3x }{3x}\cdot\frac{3x}{\sin 3x}$$

(note you should actually put limits in front of these expressions, since you've already applied the law once to get to this point)

Apply the law again, and you should get the answer. If you've never heard of Taylor series before, chances are you're not supposed to use it. This limit law is from precalc, and Taylor series is something some people encounter in their second calc course.

 

[iainfs]

Ah, OK then. I'm from the UK and we do things in a slight different order!

Do you know the small angle approximations?

$$\sin\theta \approx \tan\theta \approx \theta$$ for small $$\theta$$ in radians.

 

[karlee]

Question: sin(4x)/tan(3x) (lim approaching 0)
1- sin(4x)/((sin(3x)/(cos(3x)) (because tan=sinx/cosx
2- (sin(4x)(cos(3x))/sin(3x) (fraction on a fraction = flip up and multiply)
3-Next you're going to add in 3/3 and 4/4, this is allowed because they are both equivalent to 1. Because you know that sin(anything)/anything = 1 and anything/sin(anything) you can separate everything like so:
(sin(4x)/4x) x (3/sin(3x)) x (cos(3x)/1) x (4/3)
Once you put in the 0 you will get
1 x 1 x 1 (cos0=1) x 4/3
= 4/3
I hope this helps/I didn't make any mistakes.