Interpreting expression with big-O notation in the exponent ($f(x) = x^{1+O(1)}$)

Question

How should one interpret the notation $f(x) = x^{1+O(1)}$? I'm a bit confused as to what this means. Does it merely suggest that f(x) grows as some integer power of x?

Answer 1

Usually it would mean that there exists a function $g(x)$, satisfying

$$ g(x) = O(1), $$

such that

$$ f(x) = x^{1+g(x)}. $$

For example,

$$ f(x) = x^{1+\sin(x+1/x)} $$

is of this form both as $x \to 0$ and as $x \to \infty$.