Take f[x] = (3 x^2 - 1)/(x^2 + x + 1). Find a good global scale picture of the graph of f[x] and explain why it looks the way it does.

Answer:

Clear[f];
f[x_]=(3 x^2 -1)/(x^2 + x +1)

Now plot f[x] over several, increasingly larger ranges for x values:

Plot[f[x],{x,-10,10}, PlotRange->All]

Plot[f[x],{x,-100,100}, PlotRange->All]

Plot[f[x],{x,-1000,1000}, PlotRange->All]

In this example we are approaching a horizontal asymptote y=3 for both very large and very large negative x values. This is not a surprise since the dominant terms of f[x] are 3 x^2 in the top and x^2 in the bottom. We expect the graph of f[x] to look more and more like the graph of y = (3 x^2)/x^2 = 3. Since the dominant terms don't change for x -> -, the asymptote is a two-sided asymptote.