<s>[INST] Why using fewer terms of Taylor series doesn't give $0/0$ but gives a wrong answer? I was reading a Calculus book and I saw this problem which looks easy:
$$\lim _{x \rightarrow 0} \frac{2 x \cos x- \sin 2x}{x^3} = ?$$
It's a 0/0 limit and it's using some of the Taylor series of $\sin$ and $\cos$ expressions to solve the problem.
I know that the First and Second way should be correct because it's using more expressions of the Taylor series around 0.
What I can't figure out is WHY using fewer expressions of the Taylor series in the Third way doesn't give 0/0 but gives a wrong answer?
First way:
$$\lim _{x \rightarrow 0} \frac{2 x \cos x-2 \sin x \cos x}{x^3}=\lim _{x \rightarrow 0} \frac{2 \cos x(x-\sin x)}{x^3}=\lim _{x \rightarrow 0} \frac{2 \cos x\left(x-x+\frac{x^3}{6}\right)}{x^3}=\lim _{x \rightarrow 0} \frac{2 \cos x\left(\frac{x^3}{6}\right)}{x^3}=\frac{1}{3}$$
Second way:
$$\lim _{x \rightarrow 0} \frac{2x(1-\frac{x^2}{2})-(2x-\frac{8x^3}{6})}{x^3}=\lim _{x \rightarrow 0} \frac{2x-x^3-2x+\frac{8x^3}{6}}{x^3}=\lim _{x \rightarrow 0} \frac{\frac{x^3}{3}}{x^3}=\frac{1}{3}$$
Third way:
$$\lim _{x \rightarrow 0} \frac{2 x \cos x- \sin 2x}{x^3} =\lim _{x \rightarrow 0} \frac{2 x \cos x-2x}{x^3}=\lim _{x \rightarrow 0} \frac{2x(\cos x -1)}{x^3}=\lim _{x \rightarrow 0} \frac{2x(-\frac{x^2}{2})}{x^3}=-1$$
 [/INST] You are dividing by $x^3$ at the end, so you need all possible terms at least of degree $3$ in the numerator to be present, otherwise you're basically guaranteed to change the value of the limit.
Let's keep the error term in the third way, and see what happens. I will do that the following way: we have
$$
\sin(2x) = 2x + x^3\cdot g(x)\\
\cos(x) = 1-\frac{x^2}2 + x^4\cdot h(x)
$$
for some functions $g$ and $h$ where $g(x)$ and $h(x)$ are bounded as $x\to 0$. (It is more common to use $O(x^3)$ instead of $x^3\cdot g(x)$ and $O(x^4)$ rather than $x^4\cdot h(x)$. But the $O$ terms can be a bit unintuitive to arithmetize with, so if you are unaccustomed to working with error terms, I think that my approach here is closer to what you are already used to.)
Then we follow the steps in your third way and see what we get:
$$
\frac{2 x \cos x- \sin 2x}{x^3} =\frac{2 x \cos x-(2x + x^3\cdot g(x))}{x^3}\\
=\frac{2x(\cos x -1) - x^3\cdot g(x)}{x^3}\\
=\frac{2x(-\frac{x^2}{2} + x^4\cdot h(x)) + x^3\cdot g(x)}{x^3}\\
=-1 + x\cdot h(x) - g(x)
$$
and we see that in order to assess the limit as $x\to 0$, we don't need to know more about $h$ (it is bounded, so $x\cdot h(x)\to 0$), but we do need to know more about $g(x)$. Of course, it is easy to go back and check that $g(x) = -\frac8{3!} + x^2\cdot g_1(x)$ for some function $g_1$ that is bounded for $x\to 0$. Which is enough to conclude that the limit is indeed $\frac13$.
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