How would I change this problem into a series and evaluate the limit?
(e^x) - 1
--------
sinx
any help greatly appreciated as I'm completely stuck.
The MacLaurin series for [tex]e^x is 1+ x+x^2/2+ x^3/3!+ ...
What do you get when you subtract 1 from that.
The MacLaurin series for sin(x) is x- x^3/3!+ x^5/5!+ .... If you divide the previous series by that, do you see that you can cancel an "x"? Then, as x goes to 0, all but the constant terms will disappear.
ok, so the first x's cancel and then as I divide. (x^2/2! divided by -x^3/3! etc...)
I wind up with:
-3x^-1 + 20x ^-2 - 210x^ -3 etc....
So 2 things:
1) is my division right? there is clearly a pattern just not sure if its the right one.
2) If it is right do I just need to take the derivative of each and calculate the limit from there or is there more i'm missing?
edit: as x goes to 0 how would all the x's dissapear. Wouldnt I just get a bunch of 0 over 0's?
Big O notation - Wikipedia, the free encyclopedia
$\displaystyle \lim_{x \to 0}\frac{x+O(x^2)}{x+O(x^3)}=\lim_{x \to 0}\frac{1+O(x)}{1+O(x^2)}=1$
Instead of messing around with all the $\displaystyle c_2x^2+c3x^3 + ... + c_nx^n$ terms, you can simply say that $\displaystyle e^x = 1+ x+O( x^2)$ with an error smaller than $\displaystyle |kx^2|$ as $\displaystyle x \to 0$, where $\displaystyle k$ is a constant. In particular, $\displaystyle \lim_{x \to 0}O(x^n)=0$ for all $\displaystyle n > 0$.
It's basically the same thing. I have just truncated the Maclaurin series for $\displaystyle e^x$ and $\displaystyle \sin x$ a bit so you don't have to write out all the terms. You can of course still write it like this:
$\displaystyle \lim_{x \to 0}\frac{(1 + x + \frac{x^2}{2!} + ... + \frac{x^{n}}{n!})-1}{x+\frac{x^3}{3!}+...+\frac{x^{2n+1}}{(2n+1)!}}= \lim_{x \to 0}\frac{1 + \frac{x}{2!} + ... + \frac{x^{n-1}}{n!}}{1+\frac{x^2}{3!}+...+\frac{x^{2n}}{(2n+1) !}}=1$
1st off thanks for all the help.
2nd, and this is the same question I asked the other guy..
why do all the x terms vanish. As x approaches 0 wouldnt we be stuck with a bunch of 0 over 0's?
I know the limit is 1 because I already did it with l' hospital but I just dont understand how the x's vanish and why the series does not wind up in an inderteminate form.