Leading order - Knowlpedia

If a function has leading term $a_k\phi_k$ in an asymptotic expansion, its leading order is the comparison scale $\phi_k$ , up to a nonzero constant factor. Thus a leading term $3\varepsilon^2$ has order $\varepsilon^2$ . The limiting variable and the chosen scale are part of the statement.

Cancellations change the order

If two quantities of order $\varepsilon$ have equal leading coefficients, their difference may have order $\varepsilon^2$ or smaller. An upper bound $f=O(\varepsilon)$ alone does not establish a nonzero leading term.