What is the difference between $\lim_{h\to0}\frac{0}{h}$ and $\lim_{h\to\infty}\frac{0}{h}$?

What is the difference (if any) between-

$$\lim_{h\to0}\frac{0}{h} \text{ and } \lim_{h\to\infty}\frac{0}{h}$$

I argue that both must be $=0$ since the numerator is exactly $0$. But one fellow refuses to agree and argues that the first limit can't be $0$ as anything finite by something tending to $0$ is always $\infty$.

So,how can I explain it to the person? Also, if possible can anyone provide some good reference on this particular issue (Apostol perhaps)?

Thanks for any help!

• Graph the function $f(x) = \dfrac{0}{x}$. Perhaps by seeing it, it will make more sense to the 'fellow'. Explain the geometric interpretation of a limit. – InterstellarProbe Aug 21 '18 at 17:27
• "Anything finite by something tending to 0 is always $\infty$" is not true (at least as long as $0$ counts as finite) -- as this very example shows. Case closed. – Henning Makholm Aug 21 '18 at 17:41
• @tatan: If they don't want to consider your arguments, you can't. Give it up; life is too short for some things. – Henning Makholm Aug 21 '18 at 17:43
• Let "the fellow" read this. – drhab Aug 21 '18 at 18:09
• It's not that the numerator is zero, it's that the fraction is zero. Does your "fellow" agree that $\lim_{h\to0}0=0$. If he does, but doesn't then agree that $\lim_{h\to0}0/h=0$, then there is little hope for him. – Lord Shark the Unknown Aug 21 '18 at 18:22

In both cases you are dividing zero by a nonzero number .

Thus your fraction is identically zero and as a result the limit is zero.

For the first one let me make an intuitive explanation of what's going on with $\lim_{h\to 0}\frac{0}{h}$.

The above limit means "What value does the above expression get while $h$ gets arbitrarily close to $0$", meaning that $h$ is number very close to $0$ but not equal to that. Suppose $h=0.000001$. Then $\frac{0}{h}=\frac{0}{0.000001}=0$. Even if this number gets even closer to $0$ you see that the expression continuous to be equal to $0$. That's because $0$ divided by any number obsiously gives $0$ as the answer. Hence $\lim_{h\to 0}\frac{0}{h}=0$.

I wish I helped!

When in doubt take it back to the definition.

$\lim_\limits{x\to 0} f(x) = 0$ means

$\forall \epsilon >0, \exists \delta >0 : 0<|x|<\delta \implies |f(x)|<\epsilon$

As $f(x) = 0$ for all $x \ne 0$ the definition above is satisfied.