I was doing a GRE mock exam today and found a debatable question in the Quantitative portion. Question*
The question compares the two functions: for $$x>1$$ which one is larger
$$\lim _{x\to \infty }\left(\frac{x}{\left(x+1\right)}\right)$$ $$\lim _{x\to \infty }\left(\frac{-x}{\left(1-x\right)}\right)$$
Which for most part the second function (or Quantity B) is greater than the first function (or Quantity A)*
However, the two reaches 1 as x approches $\infty$. $$\lim_{x \to \infty} \frac{x}{x + 1}=\lim_{x \to \infty} \frac{-x}{1-x}=1\\$$
The question had "Quantity B is greater" as an answer which isn't true under every cases of x (in this case $\infty$). Therefore the most correct answer this case should be "The relationship cannot be determined from the information given".
My question is: am I missing something here? Is there a special rule in this case mathematically speaking?
*: Stack Exchange doesn't allow me to attach image before 10 reputations
The question mentions that $x>1$ which implies that the domain of the function is $x \in \mathbb{R}, x>1$.
In the image you have attached, you can clearly see that the second function is always greater than the first for all $x>1$.
By definition, the values become equal at $\infty$ which is not a part of the real numbers and hence, the first function is always smaller than the second in the common domain defined for the two functions.