# Solving limit using tailor series

1. Mar 9, 2009

### transgalactic

$$\lim _{x->0} \frac{cos(xe^x)-cos(xe^{-x})}{x^3}\\$$
$$e^x=1+x+O(x^2)\\$$
$$e^{-x}=1-x+O(x^2)\\$$
$$xe^x=x+O(x^2)\\$$
$$cos(x)=1-\frac{1}{2!}x^2+O(x^3)\\$$
$$\lim_{x->0} \frac{1-\frac{1}{2!}(x+O(x^2))^2+O(x^3)-1+\frac{1}{2!}(x+O(x^2))^2+O(x^3)}{x^3}$$

but i dont know how to deal with the remainders
there squaring of them etc..

??

2. Mar 9, 2009

### tiny-tim

Hi transgalactic!

hmm … from the x3 on the bottom, I'd guess you need to specify the x2 terms as well, and not start the Os until O(x3)

or you could just use cosA - cosB = 2.sin(A+B)/2.sin(A-B)/2

3. Mar 9, 2009

### transgalactic

i substituted functions one into another
that what i got.
where is my mathematical mistake there??

how to open this expression and having one O()
??

4. Mar 10, 2009

### transgalactic

i tried to solve it again
$$\lim _{x->0} \frac{cos(xe^x)-cos(xe^{-x})}{x^3}\\ e^x=1+x+O(x^2)\\ [tex] e^{-x}=1-x+O(x^2)\\$$
$$xe^x=x+x^2+O(x^2)$$
$$xe^{-x}=x-x^2+O(x^2)$$
$$cos(x)=1-\frac{1}{2!}x^2+O(x^3)\\$$
$$\lim_{x->0} \frac{1-\frac{1}{2!}(x+x^2+O(x^2))^2+O(x^3)-1+\frac{1}{2!}(x-x^2+O(x^2))^2+O(x^3)}{x^3}=\\$$
$$=\lim_{x->0} \frac{1-\frac{1}{2!}(x^2+O(x^2))+O(x^3)-1+\frac{1}{2!}(x^2+O(x^2))+O(x^3)}{x^3}=0\\$$

why i got 0??

5. Mar 10, 2009

### tiny-tim

Hi transgalactic!

Thta's actually pretty good, except …
should be O(x4) at the end, not O(x3)

and in the last line you should have x3 terms also …

they're the ones that don't cancel!

(and btw, it's "taylor", and why do you keep writing 2! instead of just 2? )