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Say if I have some expression $\frac{h}{k^2}$ or $\frac{h^2}{k}$ and I send both $h,k \rightarrow 0$. Can I say that the expressions tend to zero without further explanation?

When we send these constants to zero are we sending them to zero at the same speed? Or even assuming similar initial conditions? E.g if in the first expression we were to assume that $k^2 < h \implies \frac{1}{k^2} > \frac{1}{h}$ and hence $$\frac{h}{k^2}>\frac{h}{h}=1$$

This leads me to believe that we must provide a further explanation. i.e that the restriction $k^2 > h$ must hold in order for $\frac{h}{k^2}\rightarrow 0$

Am I thinking about this correctly or am I looking at this the wrong way?

  • Yes, you need to say something about the relative rates in order to understand limits of functions of both variables, in general. – Ian Nov 30 '16 at 12:46
  • $f(h,k)/g(h,k)$ can have any limit as $h,k \to 0,0,$ without knowing what $f,g$ are. – coffeemath Nov 30 '16 at 12:47
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    Same rate h = k, what does that give you? how about $h = k^2$ or $h=k^{\frac{1}{2}}$? – Paul Nov 30 '16 at 12:51
  • I see, so in order to say that the first statement converges as the variables tend to zero I would have to mention the conditions that result in convergence. This makes sense. Thanks as always! –  Nov 30 '16 at 12:55

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