Simple evaluation of function

Question

Let $F(x, y) = (x^{-\alpha} + y^{-\alpha} - 1)^{-1/\alpha}$ for some $\alpha > 0$. Show that $$F(0, y) = 0$$ According to my textbook this is trivial, but I really don't understand. How do I evaluate $0^{-\alpha}$ for example? Any hints would be appreciated.

Answer 1

So, rewriting this slightly, you have $$F(x,y)=\frac{1}{(\frac{1}{x^\alpha}+\frac{1}{y^\alpha}-1)^{1/\alpha}}$$

As $\alpha>0$, as $x\to 0$, $$F(x,y)\to \frac{1}{\infty}=0$$ since $\frac{1}{x}\to \infty$ and $\frac{1}{N}$ tends to 0 as $N\to\infty$

So, being a little lazy your $0^{-\alpha}=\infty$ and $1/\infty=0$