I'd been reading about indeterminate forms for some time, so it became quite intuitive to me that $$(\rightarrow 0 )\cdot( \rightarrow \infty), \quad (1)$$ is indeterminate.

But then I came across this $$\frac {\rightarrow 0}{\rightarrow \infty}, \quad (2)$$ which my book tells me is determinate and is equal to $(\rightarrow 0)$. To me, it seems that the intuition I developed to explain myself why $(1)$ is determinate tells me that $(2)$ can't be determinate.

Can someone give an intuitive explanation as to why these two forms/expressions are indeterminate and determinate, respectively?


The intuition is that if you take something small, and divide by something large, you always get something small.

As opposed to if you take something small and multiply by something large. That could be either small, large, or in-between, depending on how small the small thing is and how large the large thing is. This is what makes $(\to0)\cdot(\to\infty)$ indeterminate.

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