Uncovering the Mystery of an Unlikely Limit Result

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The limit \lim_{x\to\infty} \sqrt{x^2+x} - x = \frac{1}{2} challenges the assumption that leading terms dominate in polynomial sequences. The discussion highlights that while \sqrt{x^2+x} approaches x as x becomes large, it actually approximates to x + \frac{1}{2} when considering finer approximations. The error in the approximation \sqrt{x^2+x} - x is asymptotically on the order of 1/x, which allows for the limit to be determined accurately. Participants emphasize the importance of changing the expression's form to evaluate limits correctly, particularly when the degrees of the numerator and denominator are the same. Understanding these nuances in asymptotic behavior is key to resolving such limit evaluations.
  • #31
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?

You have to use the fact that f(x) is differentiable as well. Take the function f(x)=x^{1/3}. All values of this function are finite, but the function is not less than C|x| for any C whatsoever when x is sufficiently close to 0.
 

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