I am trying to proof from the defition of big-O that $$ \lim_{x \to 0} \frac{O(|x|^2)}{1+O(|x|^4)} = 0. $$ It seems not that hard but It can't quite get there. I can only bound the numerator to get something like $$ \frac{O(|x|^2)}{1+O(|x|^4)} \leq \frac{C|x|^2}{1+O(|x|^4)} = \frac{C}{1/|x|^2+O(|x|^2)} $$ I don't know how to continue from here.
2026-04-08 12:51:32.1775652692
Limit of big O term.
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The numerator goes to $0$. The denominator goes to $1$. So...