# Thread: Finding the limit

1. ## Finding the limit

okay

so I am trying to find the limit. LIM sin (squared) 3x / x (squared) x--> 0

I am soo unsure of how to do this. Any suggestions?? btw we havent gone over the L'H rule yet...

2. Originally Posted by MZTakara
okay

so I am trying to find the limit. LIM sin (squared) 3x / x (squared) x--> 0

I am soo unsure of how to do this. Any suggestions?? btw we havent gone over the L'H rule yet...
$\displaystyle \displaystyle\lim_{x\to 0}\frac{\sin^2(3x)}{x^2}=\lim_{x\to 0}\frac{9(\sin^2(3x))}{(3x)^2}=9\cdot\lim_{x\to 0}\left(\frac{(\sin(3x))}{(3x)}\right)^2=\cdots$

3. For small values of x, $\displaystyle \sin x \approx x$.

4. Use this guy

$\displaystyle \displaystyle \lim_{x\to 0} \frac{\sin x }{x} = 1$

Make

$\displaystyle \displaystyle \frac{\sin^2 3x }{x} = \frac{\sin 3x }{x}\times \frac{3}{3}\times \frac{\sin 3x }{1}$

5. but sin is squared....that does not matter???

6. Originally Posted by dwsmith
$\displaystyle \displaystyle\lim_{x\to 0}\frac{\sin^2(3x)}{x^2}=\lim_{x\to 0}\frac{9(\sin^2(3x))}{(3x)^2}=9\cdot\lim_{x\to 0}\left(\frac{(\sin(3x))}{(3x)}\right)^2=\cdots$
$\displaystyle \displaystyle9\cdot\lim_{x\to 0}\left(\frac{(\sin(3x))}{(3x)}\right)^2=\cdots$

Let 3x = t

$\displaystyle \displaystyle\lim_{t\to 0}\left(\frac{(\sin(t))}{(t)}\right)^2=\cdots$

What is 1 squared?

7. one....>_>

Thanks

8. Originally Posted by MZTakara
but sin is squared....that does not matter???
No. $\displaystyle \sin^{2}(3x) = \sin(3x)* \sin(3x) \approx 3x*3x = 9x^{2}$ for small values of x.

9. lol..ima so fail this semester.

10. Originally Posted by MZTakara
lol..ima so fail this semester.
Not when you've got MHF to turn to when you're feeling low :P