# What interval is F(x) = (x^2)/(x^2+3) increasing, decreasing?

Dec 7, 2016

#### Answer:

$F \left(x\right)$ is increasing for $x \in \left(0 , + \infty\right)$ and decreasing for $x \in \left(- \infty , 0\right)$

#### Explanation:

$F \left(x\right) = {x}^{2} / \left({x}^{2} + 3\right)$

$= \frac{1}{1 + \frac{3}{x} ^ 2}$

Hence: $L i {m}_{\text{x->+oo}} F \left(x\right) = 1$ and $L i {m}_{\text{x->-oo}} F \left(x\right) = 1$

Also notice, $F \left(0\right) = 0$ which is an absolute minimum for $F \left(x\right)$

Therefore:
$F \left(x\right)$ decreases from 1 for $x < 0$ and increases to 1 for $x > 0$

This can be seen from the graph of $F \left(x\right)$ below:

graph{x^2/(x^2+3) [-7.025, 7.02, -3.51, 3.51]}