Is $\infty \times 0$ just $\frac00$?

Question

When I solve some limit I get infinity times zero in the answer, but isn't infinity just $\frac10$ and $\frac10 \times 0 = \frac00$. Can I just use L'Hospital's rule there?

Answer 1

Can I just use l'Hospitals rule there?

If direct substitution into a limit gives you $\infty \cdot 0$ then you can use l'Hopital's rule but it's a requirement that you must first modify the expression so that direct substitution gives you $0/0$ or $\infty/\infty$.

For a very simple example, consider $\displaystyle\lim_{x \to +\infty} x e^{-x}$. Direct substitution gives $\infty \cdot 0$. But in order to use l'Hopital's rule you must first rewrite: $$ \lim_{x\to+\infty} xe^{-x} = \lim_{x\to+\infty} \frac x{e^x}$$ Now direct substitution gives you $\infty/\infty$, so now you can use l'Hopital's rule: $$ \lim_{x\to+\infty} xe^{-x} = \lim_{x\to+\infty} \frac x{e^x} = \lim_{x\to+\infty} \frac1{e^x} = 0$$