Is $\frac{y-2}{x^2+(y-2)^2}$ bounded?

Question

Is $$\frac{y-2}{x^2+(y-2)^2}$$ bounded between $[-1,1]$?

---

I do not think so, but I am not able to prove that. Maybe with a counterexample?

Thanks!

Answer 1

We need to check

$$-1\le \frac{y-2}{x^2+(y-2)^2}\le 1 $$

and the answer depends upon the domain we are considering for $(x,y)$.

Without any constraint we can see for example that for $x=0$ and $y=3/2$ we have

$$\frac{y-2}{x^2+(y-2)^2}=\frac{-1/2}{1/4}=-2 $$

and therefore the expression is not bounded.

Answer 2

Take $x=0$ so your expression is $1/(y-2)$. If $y=2$ this is undefined, and if $y$ is near $2$ (but not equal) it is very large (positive or negative).