I asked a question regarding why parabolas' arms become parallel . I received many good answer. But sometime back a question popped up in my mind- If slope of the graph of a quadratic polynomial becomes infinite as $x \to \infty$ then why do we still call it a function?
One of the main aspect of a function is that for every argument, it returns only one value. But as $x \to \infty$ the the arms of parabola become parallel i.e. it returns many values for one value of $x$. So, why do we still call it a function?
This is not quite right. Consider the parabola $f(x)=x^2$. Although the slope $f'(x)=2x$ goes to $\infty$ as $x\to\infty$, that does not mean that $f$ returns many values for a given value of $x$. For each $x\in\mathbb{R}$, $f(x) = x^2$.
However, $\infty\not\in\mathbb{R}$. You cannot take $f(\infty)$.