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I was working on getting intuition behind limits in multivariate calculus and I ran into this article.

I am mostly concerned with the case where we have functions of two or three variables. Unfortunately I do not have the necessary background to understand the proof provided but I think that if the partial derivative in a given direction is not zero in the neighborhood of the limit point (for which the numerator and denominator are zero) then we have:

$$\lim_{(x,y)\rightarrow (a,b)} \frac{f(x,y)}{g(x,y)}=\lim_{(x,y) \rightarrow (a,b)} \frac{D_vf(x,y)}{D_vg(x,y)}$$

So when seeking to resolve a question about limits (when the numerator and denominator are both zero at the point), I should quickly check the partial derivative of the numerator and denominator in convenient directions and ensure that they do not both vanish in the neighborhood. Is this a correct interpretation?

Ivo Terek
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recmath
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1 Answers1

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If the limit of the quotient exists AND the partial derivatives exist in a neighborhood AND the limit of the partial derivatives exists, then you can find the limit by the procedure you describe. Because then everything reduces to the one-dimensional case by $$\tilde{f}(t):=f((x,y)+tv)\quad \tilde{g}(t):=g((x,y)+tv).$$

However, for $f(x,y)=x+y$ and $g(x,y)=x^2+y$ the limit of $f/g$ doesn't exist if $(x,y)\to (0,0)$, although taking the directional derivative in the $v=(0,1)$ direction would yield a result $1$.

Peter Franek
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