# Finding out a limit using Taylor series.

So the limit is the following:

$$\lim_{x \to 0}{\frac{x^2-\frac{x^6}{2}-x^2 \cos (x^2)}{\sin (x^{10})}}$$

Expansions for $\sin(x)$ and $\cos(x)$ are given:

$$\sin x = x-\frac{x^3}{3!} + \frac{x^5}{5!}-...+(-1)^{n-1}\frac{x^{2n-1}}{(2n-1)!} + o(x^{2n})$$ $$\cos x = 1-\frac{x^2}{2!}+\frac{x^4}{4!}-...+(-1)^n\frac{x^{2n}}{(2n)!}+o(x^{2n+1})$$

Here is what I tried: $$\lim_{x->0}{\frac{x^2-\frac{x^6}{2}-x^2(1-\frac{x^4}{2}+\frac{x^8}{4!}+o(x^{2*5}))}{x^{10}+o(x^{10*2})}}=\lim_{x->0}{\frac{-\frac{x^{10}}{4!}-o(x^{12})}{x^{10}+o(x^{20})}}$$

This is where I am stuck. I figured that the problem occurs when expending $o()$. What am I missing here?

• Also better to use \sin than just sin – Alex Apr 26 '16 at 21:35

How about factoring by $x^{10}$?

• What does $o(x^{20})/x^{10}$ tend to? – khajvah Apr 26 '16 at 21:23
• $\frac{o(x^m)}{x^n}$ tends to $0$ in $0$ for $m \geq n \in \mathbb{N}$. Go back to the definition of $o$ to see it – Vincent Apr 26 '16 at 21:26

The numerator is $$x^2-\frac{x^6}2-x^2\Bigl(1-\frac{x^4}2+\frac{x^8}{24}+o(x^8)\Bigr)=-\frac{x^{10}}{24}+o(x^{10}),$$ hence $$\frac{x^2-\cfrac{x^6}{2}-x^2 \cos (x^2)}{\sin (x^{10})}\sim_0\frac{-\dfrac{x^{10}}{24}}{x^{10}}=-\frac1{24}.$$

*Editors Note: Sorry, can't leave comments yet. When you factor in the $-x^2$ you'll end with $-\frac{1}{24}$.

• Sure, but where is there a term in $\frac1{x^4}$? – Bernard Apr 26 '16 at 21:36
• My apologies, I got the term incorrect – Alex Apr 26 '16 at 21:38

Another way is to note that $\frac{\sin x^{10}}{x^{10}} \to_x 1$, so you get your $x^{10}$ in the denominator.

• The problem still remains in numerator. $o(x^{12})/x^{10}$ doesn't give anything, right? – khajvah Apr 26 '16 at 21:27
• $\frac{x^{12}}{x^{10}} = x^2 \to 0$ – Alex Apr 26 '16 at 21:28
• Oh god, completely missed that $x->0$... Thanks for help and sorry for wasting your time :) – khajvah Apr 26 '16 at 21:29

$$\lim_{x\to0}\frac{x^2-\frac{x^6}{2}-x^2\cos(x^2)}{\sin(x^{10})} = \lim_{x\to0}\frac{x^2-\frac{x^6}{2}-x^2(1-\frac{x^4}{2}+\frac{x^8}{24}-\frac{x^{12}}{720}+\cdots)}{x^{10}-\frac{x^{30}}{6}+\cdots}$$

$$=\lim_{x\to0}\frac{x^2-\frac{x^6}{2}+(-x^2+\frac{x^6}{2}-\frac{x^{10}}{24}+\frac{x^{12}}{720}-\cdots)}{x^{10}-\frac{x^{30}}{6}+\cdots}$$

Canceling terms in the numerator simplifies the equation to:

$$\lim_{x\to0}\frac{-\frac{x^{10}}{24}+\cdots}{x^{10}-\cdots}$$

The other terms in the series are unnecessary, so

$$\lim_{x\to0}(-\frac{x^{10}}{24}*\frac{1}{x^{10}}) = -\frac{1}{24}$$