<s>[INST] Why can the Taylor formula be used to evaluate the limit,Why does its remainder converge to zero？ A simple example $\lim\limits_{x\to0}\frac{\cos(x)-1}{x^2}=\frac{1-\frac{1}{2}x^2+o(x^2)-1}{x^2}=-\frac{1}{2}$

If taken out alone, its limit is zero.How can I judge that the sum of infinite limits converges to zero.
Such as:$\lim\limits_{x\to0}\frac{o(x^2)}{x^2}=\lim\limits_{n\to\infty}[\sum\limits_{i=3}^{n} (\lim\limits_{x\to0}\frac{x^i}{x^2})]=?$
Clarification of the problem
For example $\lim\limits_{n\to\infty}(\frac{1}{n}+\frac{1}{n}+.....+\frac{1}{n})=\lim\limits_{n\to\infty} n*\frac{1}{n}=1$
In the case of an infinite number of items,I cannot simply think of adding an infinite number of zero
 [/INST] Apply the rules of asymptotic analysis:
$$\frac{\cos x-1}{x^2}=\frac{1-\frac{1}{2}x^2+o(x^2)-1}{x^2}=-\frac 12+o(1).$$
Added: an elementary way to determine the limit:
$$\frac{\cos x-1}{x^2}=\frac{\cos^2 x-1}{x^2(\cos x +1)}=-\underbrace{\frac{\sin^2x}{x^2}}_{\substack{\downarrow\\1^2=1}}\,\underbrace{\frac1{\cos x+1}}_{\substack{\downarrow\\1+1=2}}$$
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