Bounds for $\frac{x-y}{x+y}$ How can I find upper and lower bounds for $\displaystyle\frac{x-y}{x+y}$? So I do see that
$$\frac{x-y}{x+y} = \frac{1}{x+y}\cdot(x-y) = \frac{x}{x+y} - \frac{y}{x+y} > \frac{1}{x+y} - \frac{1}{x+y} = \frac{0}{x+y} = 0$$
(is it correct?) but I don't get how to find the upper bound.

so we have $f(x,y)=\frac{x-y}{x+y}$, we want to show that $f(x,y)$ is unbounded, therefore we fix $y=y_0>0$ and then have
$$
f(x,y_0)=f(x)=\frac{x-y_0}{x+y_0}
$$
now choose $x=x^*-y_0$ which gives us
$$
f(x)=\frac{x^*-2y_0}{x^*}
$$
and now observe that for $\lim_{x^*\to0+}\frac{x^*-2y_0}{x^*}=-\infty$ and for $\lim_{x^*\to0-}\frac{x^*-2y_0}{x^*}=\infty$.
So $f(x,y)$ is indeed unbounded as we thought.
