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We can talk about how "quickly" an infinite series approaches $0$ by talking about an asymptotic bound on its terms - a series that is $O(1/x)$ converges more slowly than one that is $O(1/x^2)$, etc.

I am confused about what we mean when we say that a function rapidly approaches a limit at a (finite) point. For example, consider the functions $x$ and $x^2$ as $x$ approaches $0$. I think we say that $x^2$ approaches $0$ "more quickly" than $x$ does as $x\to 0$ because $x^2$ is $o(x)$ as $x\to 0$, so $x^2$ reaches any arbitrary "closeness to $0$" before $x$ does as $x\to 0$.

On the other hand, close to $0$, it also seems that $x$ is approaching $0$ "more quickly" than $x^2$ in the sense that the magnitude of derivative of $x$ is larger than that of $x^2$.

So my question is which one of these does "more quickly" generally mean in informal math-language, and if anyone has any insight into the intuition here.

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$x^2$ approaches 0 "more quickly" than $x$ because $\lim_{x\rightarrow 0^+}{\frac{x}{x^2}}=+\infty$. That is, the ratio of $x$ to $x^2$ gets arbitrarily large. More generally, you can say that $f(x)$ approaches $x_0$ "more quickly" than $g(x)$ does if $\left|\lim_{x\rightarrow x_0}{\frac{f(x)}{g(x)}}\right|<1$. – Hayden Mar 20 '14 at 18:16
Note that L'Hopital means that the [$x^2=o(x)$ as $x\to 0$] definition and the derivative definition are equivalent. – Alyosha Mar 20 '14 at 18:18
up vote 2 down vote accepted

Notice that you are not using the size of the derivative in discussing limits at $\infty.$ So, you don't emphasize the derivative in limits at $0.$ You just talk about which function gets close to $0$ earlier, which at $0^+$ means for larger $x.$ Also, at $\infty^+,$ the derivative information (in absolute value) agrees with the size information; note that we are thinking of a point moving to the right along the $x$ axis.

I suppose it is fair to say that the derivative information is backwards of this, at $0^+, $ and this is happening because approaching $0^+$ we are moving to the left. The slowest approach to $0$ you are likely to see is Holder continuity such as $\sqrt x,$ where the derivative is actually infinite at the origin. `Slow' meaning: you need to have $x < 0.0001$ to get $\sqrt x < 0.01$

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You could slow the convergence even more by considering $\sqrt[n]{x}$ for larger $n$. – Baby Dragon Mar 20 '14 at 18:30
yes; still Holder – Will Jagy Mar 20 '14 at 18:31

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