# Trigonometry Limit

1. Oct 29, 2015

### songoku

1. The problem statement, all variables and given/known data
Find the value of:

lim x approaches 0 of : (3x - sin 3x) / (x2 sin x)

2. Relevant equations
Trigonometry identity
Limit properties
No L'hopital rule

3. The attempt at a solution
I tried changing sin 3x to -4sin3x + 3 sin x but then I stuck. Is changing sin 3x correct way to start solving the question?

Thanks

2. Oct 29, 2015

### Svein

You do remember that $\lim_{z\rightarrow 0}\frac{\sin(z)}{z}=1$?

3. Oct 29, 2015

### songoku

Yes I do but I don't know how to use it to solve this question.

Dividing all the terms by x resulting in:

$\lim_{x\rightarrow 0} {\frac{3 - \frac{\sin(3x)}{x}}{x \sin(x)}}$

$= \lim_{x\rightarrow 0} \frac{3 - 3}{x \sin(x)}$

Then stuck

4. Oct 29, 2015

### haruspex

That step is not valid. You cannot take the limit in the numerator only, then in the denominator. The two must be done together.
I would expand sin() as a power series, keeping two or three terms. Not sure if that would be considered allowable in your context.

5. Oct 30, 2015

### songoku

Sorry that is not allowed. No other way to solve it?

6. Oct 30, 2015

### Staff: Mentor

What's the context for this problem? By that, I mean where did you see this problem? The limit is apparently 4.5, but the only way I've been able to get that is by using haruspex's suggestion, in addition to using Excel to approximate the limit.

7. Oct 30, 2015

### SammyS

Staff Emeritus
It does look like your initial approach can get you part way there.

$\displaystyle \ \frac {3x-\sin(3x)}{x^2 \sin(x)} = \frac {3x-3\sin(x)+4\sin^3(x)}{x^2 \sin(x)} \$

$\displaystyle \ =\frac {3x-3\sin(x)}{x^2 \sin(x)} + \frac {4\sin^3(x)}{x^2 \sin(x)} \$​

The limit of the second term is straight forward.

The first term remains somewhat a problem. Maybe Mark or haruspex has an idea for that.

Last edited: Oct 31, 2015
8. Oct 31, 2015

### songoku

From a book I use in high school. This is the question from exercise in the book. The question says: find the limit of the following, then there are a lot of limit questions, from (a) to (z). One of the question is exactly as I posted. The book doesn't cover about power series and at that point (when I saw that question), I haven't learn about L'hopital rule yet.

Maybe the question is misplaced, should not be on that part of exercise. I should cover L'hopital rule or power series first before solving that type of question.

Thanks a lot for all the help

9. Oct 31, 2015

### Staff: Mentor

Regarding the first term of what Sammy shows above, the only techniques that I can think of are 1) expanding the sin(3x) term (which would be $3x - \frac{3x^3}{3!}$ plus terms of degree 5 and higher), or 2) using L'Hopital's rule, which has to be applied four times.. Either way gives the result I showed in my earlier post.