# How do i prove that odd roots cross the x axis and even roots don’t?

I would like to be able to explain the behavior of functions near their roots. I have found several sources claiming that one variable functions (i am not sure if this can be extended to multi variable functions) cross the x axis when the root in question has odd multiplicity but not when the multiplicity is even. Also, it looks like increasing the multiplicity of the roots affect the shape of the function near the root as it looks “flatter”. I think that this can be proven for a neighborhood of the root in question and it is probably related to the second derivative test, if f”(x)=0 then it is an inflection point isn’t it? But i don’t quite get why it looks flatter near that point.
Is there a relationship between the function appearance and the multiplicity of the roots? Any guidance would be appreciated.

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Consider limx→∞f(x)\lim_{x \to \infty} f(x) and \lim_{x \to -\infty} f(x)\lim_{x \to -\infty} f(x).
– Théophile
Oct 20 at 18:51

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say f(x)=(x-a)^ng(x)f(x)=(x-a)^ng(x), where g(x)\neq 0g(x)\neq 0. Suppose x=a\pm \epsilonx=a\pm \epsilon for small \epsilon\epsilon, and note that for continuous gg, the sign won’t change near aa.
– lulu
Oct 20 at 18:59

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