Limits at or with infinity

[math]\displaystyle{ \lim_{x \ \to \ \infty} x^2 = \infty }[/math] because the function can grow indefinitely. [math]\displaystyle{ \lim_{x \ \to \ \infty} \frac{1}{x} = 0 }[/math] because we are dividing a number by something very large, or in infinitely many small parts. Is there a rigorous way to prove that our intuition is correct in both cases? Yes, there is. This idea is pretty abstract because the whole concept is "there is a number that is very large, then we can add one and make it even larger and continue this process indefinitely".