Limits at or with infinity: Difference between revisions
(Created page with "<math>\lim_{x \ \to \ \infty} x^2 = \infty</math> because the function can grow indefinitely. <math>\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...") |
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<math>\lim_{x \ \to \ \infty} x^2 = \infty</math> because the function can grow indefinitely. <math>\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"''. | <math>\lim_{x \ \to \ \infty} x^2 = \infty</math> because the function can grow indefinitely. <math>\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"''. The other side of the same coin is to divide a number and repeat it indefinitely. We know that to divide anything in half yields smaller parts. Doing it again and again would come to an end at the atomic and then, subatomic level. But mathematics allow us to go beyond the smallest atom and there is the abstraction, how can something be smaller than a subatomic particle? | ||
Revision as of 22:56, 15 February 2022
[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". The other side of the same coin is to divide a number and repeat it indefinitely. We know that to divide anything in half yields smaller parts. Doing it again and again would come to an end at the atomic and then, subatomic level. But mathematics allow us to go beyond the smallest atom and there is the abstraction, how can something be smaller than a subatomic particle?
