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

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Answer 1 · 2016-12-07T12:42:00

$F(x)$ is increasing for $x in (0, +oo)$ and decreasing for $x in (-oo, 0)$

$F(x) = x^2/(x^2+3)$

$= 1/(1+3/x^2)$

Hence: $Lim_"x->+oo" F(x) = 1$ and $Lim_"x->-oo" F(x) = 1$

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

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

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

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

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