It seems that there are two uses of Bachman Landau notation out in the world, exemplified in these two circumstances
$$ 1 + (1/x) \sin(x) = 1 + o(1)\ \ \ \ \ \ \ \ \ \ 1 + 1/x = 1 + o(1) $$
In the first example, the $o(1)$ term is not necessarily always a positive value, because $\sin(x)$ fluctuates between $\pm 1$. In the second example, $1/x$ is always positive, so that not only do we have $1 + 1/x$ equal to 1 `in the limit', but also we have $1 + 1/x \geq 1$ for all $x \geq 0$.
These are two trivial examples, but they show that in some cases, we use $o(1)$ to denote the class of functions $f: \mathbf{R} \to \mathbf{R}$ such that $\lim_{x \to \infty} f(x) = 0$, and in the second case we use $o(1)$ to denote the class of positive functions $f: \mathbf{R} \to \mathbf{R}^+$ with $\lim_{x \to \infty} f(x) = 0$. Is there a standard notation to distinguish between these two classes? I'm thinking of using $o^+(1)$, but I don't want to use a new notation when another notation already exists.