Example
38
[1FF]We informally state this second property
If \(n≥ 1\) then \(o\Big(x^ n+o(x^ n)\Big)=o(x^ n)\).
We rewrite it like this.
If \(f(x)=o(x^ n)\) and \(g(x)=o(x^ n+f(x))\) then \(g(x)=o(x^ n)\).
We note that, for \(x≠ 0\) small, \(x^ n+f(x)\) is not zero, as there is a neighborhood in which \(|f(x)|≤ |x^ n/2|\). As a hypothesis we have that \(\lim _{x→ 0}f(x)x^{-n}=0\) and \(\lim _{x→ 0}g(x)/(x^ n+f(x))=0\) then
\[ \lim _{x→ 0}\frac{g(x)}{x^{n}}= \lim _{x→ 0}\frac{g(x)}{x^ n+f(x)}\frac{x^ n+f(x)}{x^{n}} \]
but
\[ \lim _{x→ 0}\frac{g(x)}{x^ n+f(x)}=0 \]
while
\[ \lim _{x→ 0}\frac{x^ n+f(x)}{x^{n}}=1\quad . \]