I'm still confused. I understand that 1/x -> 0 as x -> inf. But why doesn't a term like 5x -> inf at the same time, especially if 1/x and 5x are in the same equation? It's like saying 1/x + 5x -> 5x for x -> inf instead of 1/x + 5x being undefined.

If you think about what "approach" means in mathematics, it is a little deceptive. When we say "x approaches a," where a is a real number, we mean that x APPROXIMATES BUT DOES NOT EQUAL a. When, however, we say "x approaches infinity" or "x approaches negative infinity," we can't mean the same thing because no finite number approximates infinity. Instead we mean "x is a finite number with a very large absolute value."

OK. So when we say

\(\displaystyle \text {As x approaches infinity, } \dfrac{1}{x} + 5x \text { approaches } 5x\),

what we mean is almost exactly what you said:

\(\displaystyle \text {As x gets very large, } \dfrac{1}{x} + 5x \text { ALMOST equals } 5x.\)

Standard analysis, a mathematical theory underlying calculus, makes this vocabulary precise, but if you realize that "approaching a finite number" means "approximating that finite number" but "approaching infinity" means "picking a finite number with a large absolute value," limits will make intuitive sense.