# Math Help - Limits

1. ## Limits

Hello!

Here is my equation

lim sin5x-3x
x->0 3x

I think that I'm supposed to separate the sin5x/3x and the -3x/3x but do i bring the limit in front of both or only the sin5x/3x? and would the -3x/3x just work out to be -1?
Ahh a bit lost, any help would be greatly appreciated! Thanks in advance

2. This is not an equation.

$\displaystyle \lim_{x\to 0} \left( \dfrac{sin(5x)-3x}{3x} \right) = \lim_{x\to 0} \left( \dfrac{sin(5x)}{3x} - 1 \right)$

To complete it, you'll use this formula: $\displaystyle \lim_{x\to 0} \dfrac{sin(ax)}{ax}=1$.

Can you rewrite $\dfrac{sin(5x)}{3x}$ so that the formula could be applied ?

3. Originally Posted by wassup
Hello!

Here is my equation

lim sin5x-3x
x->0 3x

I think that I'm supposed to separate the sin5x/3x and the -3x/3x but do i bring the limit in front of both or only the sin5x/3x? and would the -3x/3x just work out to be -1?
Ahh a bit lost, any help would be greatly appreciated! Thanks in advance
Since this goes to $\displaystyle \frac{0}{0}$, L'Hospital's Rule is the quickest method of evaluating this limit...

$\displaystyle \lim_{x \to 0}\frac{\sin{(5x)} - 3x}{3x} = \lim_{x \to 0}\frac{\frac{d}{dx}[\sin{(5x)} - 3x]}{\frac{d}{dx}(3x)}$

$\displaystyle = \lim_{x \to 0}\frac{5\cos{(5x)} - 3}{3}$

$\displaystyle = \frac{5 - 3}{3}$

$\displaystyle = \frac{2}{3}$.

4. Yes, but "overkill"! Like killing a mosquito with a 12 guage shot gun. I much prefer Liverpool's method.
I also like the fact that Liverpool left something to be done rather than giving the answer- but I'll thank both of you since both replies were useful.

5. Originally Posted by HallsofIvy
Yes, but "overkill"! Like killing a mosquito with a 12 guage shot gun. I much prefer Liverpool's method.
I also like the fact that Liverpool left something to be done rather than giving the answer- but I'll thank both of you since both replies were useful.
I prefer the term "brute force", and I consider it to be more effective than beating around the bush. It's like punching someone in the face or talking behind their back...

6. But talking behind their back is much safer than punching them in the face!

7. What would Chuck Norris do, I wonder ?

Well, to answer the questions....

If you want to split it up

$\displaystyle\frac{Sin5x-3x}{3x}=\frac{Sin5x}{3x}-\frac{3x}{3x}=\frac{Sin5x}{3x}-1$

So, to evaluate this limit, you only need evaluate $\displaystyle\lim_{x \to 0}\frac{Sin5x}{3x}$

and then subtract 1.

An alternative method is offered by Prove It "if" you are familiar with L'Hopital's Rule.