# Math homework help

1. Mar 31, 2005

### gillgill

does this seem right?
lim (sin3x)/(x+3x^2)
x->0

=(sin3x)/[3x^2(1/3x+1)]
=1/[(1/3(0)+1)(sin3(0))]
=undefined

2. Mar 31, 2005

sin(3x) = 0
x-> 0

3. Apr 1, 2005

### Data

do you know l'Hopital's rule?

4. Apr 1, 2005

### Galileo

Can you rewrite your equation in order to use:

$$\lim_{x \to 0}\frac{\sin(3x)}{3x}=1$$?

5. Apr 1, 2005

### BobG

Don't you mean $$\lim_{x \to 0}\frac{3 \sin(x)}{3x}=1$$?

Or are you just taking advantage of the fact, that near zero, sin x = x?

L'Hopital's Rule is certainly the easiest way to solve this. If you haven't learned it yet, check the table of contents and find it. Some books teach it very early on and some wait to teach it until later on. It's easy enough to learn that I'd learn it early even if the course waits until later.

For your equation, it's simply the derivative of the top divided by the derivative of the bottom. You can do that whenever your limit winds up 0/0, or infinity/infinity.

In other words:

$$\lim_{x \to 0}\frac{\sin(3x)}{x+3x^2}=\lim_{x \to 0}\frac{3 \cos(3x)}{1+6x}$$?

Last edited: Apr 1, 2005
6. Apr 1, 2005

### HallsofIvy

Staff Emeritus
No, Galileo meant $$\lim_{x\to 0}\frac{sin(3x)}{3x}= 1$$ just like he said.

Sort of, though I wouldn't say it like that. sin x is NOT equal to x "near zero". They are only equal AT 0. What you meant to say, I am sure, is that sin x is close to x for x close to 0 and the become close as x goes to 0: $$\lim_{x \to 0}\frac{sin x}{x}= 1$$ and $$\frac{sin(3x)}{x}= \frac{3 sin(3x)}{3x}= 3\frac{sin(y)}{y}$$ where y= 3x. In other words, $$\lim_{x \to 0}\frac{sin(3x)}{x}= \lim_{x \to 0}3\frac{sin 3x}{3x}= 3\lim_{y \to 0}\frac{sin(y)}{y}= 3$$.

In my opinion, it is far simpler to use Galileo's idea than to use L'Hopital.

Last edited: Apr 1, 2005
7. Apr 1, 2005

### Data

Yes, it is. L'Hopital's result is too strong for this problem. Although it will of course get you the answer quite readily, it's a lot harder to prove than the things Galileo's method needs

Nevertheless, if he did know l'Hopital's rule, it would be the obvious path~