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If $f(x) \sim g(x)$ like for example $n! \sim \sqrt{2\pi n}(\frac{n}{e})^n$, and one wants to calculate a limit involving $f(x)$ and possibly other functions, under what circumstances is one allowed to replace $f(x)$ by $g(x)$? If for example $\frac{f(x)}{g(x)} \to 1$ but $f(x)-g(x) \to \infty$, can one still replace $f(x)$ by $g(x)$?

David
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  • If only the relative error matters, we can replace it (without making a large error, if $x$ is sufficiently large). But the absolute error will usually tend to $\infty$. So, whether $f(x)$ can be replaced depends on what exactly you want to calculate – Peter Dec 19 '16 at 13:07
  • What does relative error and absolute error mean? – David Dec 19 '16 at 13:08
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    Lets say, the actual value is $v$ and the approximation is $v'$, then the absolute error is $|v-v'|$ , whereas the relative error is $\frac{|v-v'|}{v}$ (we need $v\ne 0$ here). The relative error will tend to $0$ in the case of asymptotic equivalent functions. – Peter Dec 19 '16 at 13:11
  • Thank you. So one cannot necessarally replace $f(x)$ by $g(x)$ when calculating limits is what you're saying? – David Dec 19 '16 at 13:21
  • As you already mentioned in your question , you can use $$\lim_{x\rightarrow \infty} \frac{f(x)}{g(x)}=1$$ What other limits are you interested in ? – Peter Dec 19 '16 at 13:25
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    If you have $\frac{f(x)}{h(x)}$ or $f(x)\cdot h(x)$ and you want to calculate the limit for $x\rightarrow\infty$, you can replace $f(x)$ by $g(x)$ – Peter Dec 19 '16 at 13:28
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    Okay, thank you. – David Dec 19 '16 at 13:33

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