# How do you graph f(x)=2/x^2+1 using holes, vertical and horizontal asymptotes, x and y intercepts?

How do you graph $f\left(x\right)=\frac{2}{{x}^{2}+1}$ using holes, vertical and horizontal asymptotes, x and y intercepts?
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Trace Arias
$f\left(x\right)=\frac{2}{{x}^{2}+1}$
Since ${x}^{2}+1>0\forall x\in ℝ$ there exists no holes in f(x)
Also, $\underset{\text{x-> +-oo}}{lim}f\left(x\right)=0$
$f\prime \left(x\right)=\frac{-4x}{{\left({x}^{2}+1\right)}^{2}}$
For a maximum or minimum value; f'(x)=0
$\therefore \frac{-4x}{{\left({x}^{2}+1\right)}^{2}}=0\to x=0$
$f\left(0\right)=\frac{2}{0+1}=2$
Since $f\left(0\right)=2$ is a maximum of f(x)
The critical points of f(x) can be seen on the graph below:
graph{2/(x^2+1) [-5.55, 5.55, -2.772, 2.778]}