The limit \lim_{x\to\infty} \sqrt{x^2+x} - x = \frac{1}{2} challenges conventional logic that suggests \sqrt{x^2+x} approximates to x as x becomes large. The error arises from neglecting lower-order terms, which can be addressed through rationalization or by using the binomial theorem for better approximations. The discussion emphasizes that understanding asymptotic behavior and error analysis is crucial for evaluating limits accurately.
Students and educators in calculus, mathematicians focusing on limit evaluation, and anyone interested in deepening their understanding of asymptotic behavior in mathematical analysis.
Gib Z said:I don't really understand what I'm trying to prove..I know f is continuous, therefore all values are finite, as also shown by your theorem. Since f(x) is finite, and so is f(0), the error must also be finite. So to show that it is less than C|x| for some x we just have to choose a really large C, is that somewhat correct even if not rigorous?