Principal Part and little o notation

Question

The function is $f(x) = 2x + e^{x+x^{4/3}} -1$ and the question wants the principal part with respect to $g(x)=x$ as $x$ goes to $0^+$.

I solved it using MacLaurin expansion and I got $3x$, but should it also have the little $o$ notation with the answer, like $3x + o(x)$?

Answer 1

As $x\to 0^+$, $$e^{x+x^{4/3}}=1+(x+x^{4/3})+O(x^2)=1+x+O(x^{4/3}).$$ with Big O notation. Therefore you may write $$f(x)=3x+O(x^{4/3})\quad\mbox{or}\quad f(x)=3x+o(x^{a})$$ with any $1\leq a<4/3$.