Let $x$, $y$ be positive hyperreal numbers. Can $\frac{x}{y}+\frac{y}{x}$ be infinite? finite? infinitesimal?

Question

I'm going through problems in chapter one of Elementary Calculus: An Infinitesimal Approach by H. Jerome Keisler.

I'm doing an even numbered one and there are only answers for odd numbered problems. Q. 42, section 1.5 problems.

Let $x$, $y$ be positive hyperreal numbers. Can $\frac{x}{y}+\frac{y}{x}$ be infinite? finite? infinitesimal?

\begin{align} \frac{x}{y}+\frac{y}{x} = \frac{x^2+y^2}{xy} \\ \end{align}

In the case where $x = y$ we have

$$\frac{2x^2}{x^2} = 2$$

And this is a finite.

In the case where $x \ge y$ we have

$$\frac{x^2+y^2}{xy}$$

and the $x^2$ in the numerator will be greater than the $xy$ in the denominator.I am just not sure if the numerator is so much larger than the denominator to say this expression is infinite in this case. I don't fully understand these hyperreals I suppose.

Any help here?

Answer 1

I might be missing something, but the problem seems pretty straightforward to me.

Let $ \frac{x}{y} = t$, the required expression is now t+1/t. It is trivial that as t tends to infinite 1/t tends to 0 so the required expression can be infinite. A simple application of AM-GM inequality will show that for positive numbers the required sum cannot be infinitesimal. $$ \frac{t+1/t}{2} \ge \sqrt{t*1/t} = 1$$