Is this equality about limits True? Given that $f,g$ are two functions such that $\lim \limits_{x \to \infty} \frac{f(x)}{g(x)} = 0$, does this imply that $\lim \limits_{x \to \infty} f(x)+g(x) = \lim \limits_{x \to \infty} g(x)$ ?!
 A: That's not true.
Consider for example,
$$f(x)=1, \ \ \ \ \ \ \ g(x)= (-1)^{[x]}x$$
where $[x]$ is the greatest integer function.
A: If $\lim g(x)$ exists, then it is true. We have the following equality $$f(x)+g(x)=g(x)\left(1+\frac{f(x)}{g(x)}\right)$$
And since both $\lim g(x)$ and $\lim\frac{f(x)}{g(x)}$ exist, $\lim g(x)\left(1+\frac{f(x)}{g(x)}\right)$ also exists and, since it can't be an indeterminate form because the limit of the right part is $1$, we have
$$\lim g(x)\left(1+\frac{f(x)}{g(x)}\right)=(\lim g(x))\left(1+\lim \frac{f(x)}{g(x)}\right)=(\lim g(x))(1+0)=\lim g(x)$$
Combining this with the previous equality, we obtain
$$\lim f(x)+g(x)=\lim g(x)$$

But $\lim g(x)$ may not be defined. Take $g(x)$ to be any function for which $\lim g(x)$ is undefined, and set $f(x)=g(x)/(1+x)$.
A: Assuming that both $f$ and $g$ are real positive functions there are two cases where $\lim \limits_{x \to \infty} \frac{f(x)}{g(x)} = 0$. $\lim \limits_{x \to \infty} g(x) = \infty$ or $\lim \limits_{x \to \infty} f(x) = 0$. In both cases is trivial that the second equality holds. This arguments doesn't sostain if the function is negative or complex.
A: You have to interpret the equality of limits like $$\lim_{x \to a}f(x) = \lim_{x \to a}g(x)$$ in the following manner:
1) If one of the limits exists then the other exists and both are equal.
2) If one of the limits diverges to $\infty$ (or $-\infty$) so does the other.
3) If one of the functions oscillates finitely or infinitely as $x \to a$ then so does the other function (with exactly the same set of accumulation points for the range of both functions).
We usually combine the above three statements into one and say that functions $f$ and $g$ have the same limiting behavior as $x \to a$.
Under this wider interpretation of equality of limits, we have the following conclusion: $$\lim_{x \to \infty}\frac{f(x)}{g(x)} = 0\Rightarrow \lim_{x \to \infty}f(x) + g(x) = \lim_{x \to \infty}g(x)$$ So the answer to your question is a resounding YES!
