= Asymptotic analysis
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Asymptotic analysis describes what remains dominant as an argument approaches a specified <limit>. It compares <functions> or <sequences> by scale: how much of a complicated expression can be replaced by a simpler one, and how large is the omitted part? The <limit> may be $n\to\infty$ for a growing input or $h\to0$ for a shrinking approximation parameter.
For example, $n^2+10^5n$ is eventually dominated by its quadratic term, since division by $n^2$ gives $1+10^5/n\to1$. But at $n=100$, the linear term contributes $10^7$, compared with only $10^4$ from the quadratic term. A statement about the limiting scale and a good approximation at a particular input are different claims.
Three useful comparisons keep their strengths distinct. <Big O notation> supplies an eventual <upper bound> up to a constant multiplier; <big theta notation> supplies both an <upper bound> and a <lower bound> up to constants; <asymptotic equivalence> makes the ratio tend to one. <Little o notation> says that a term becomes negligible relative to a specified scale. These distinctions make approximations checkable instead of merely calling one expression small.
The same language connects <time complexity> with <Taylor remainders>. In an operation-count model, one asks how a cost scales with input size. In a <Taylor expansion>, one asks how the approximation error scales with the increment. Neither use requires interpreting the unspecified constant as one. The <asymptotic comparison of exponential functions> gives another example: dividing by the largest contributing exponential reveals the terms that disappear.
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