Take f[x] = Sin[x]/e^(x/100). 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_]= Sin[x]/E^(x/100)

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

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

Plot[f[x],{x,-60,-40}, PlotRange->All]

Plot[f[x],{x,-500,500}, PlotRange->All]

In this problem, the bumps in f[x] seem to be increasing in size as
x -> -. This is not a surprise---as x -> -, the term in the demominator, e^(x/100) goes to 0. As a result, overall the size of f[x] is increasing.
What happens as x -> is quite different. In this problem it appears that we have y=0 as a horizontal asymptote as x -> . Let's make a few more plots, concentrating on what's happening for
x > 0:

Plot[f[x],{x,0,10}, PlotRange->All]

Plot[f[x],{x,0,100}, PlotRange->All]

Plot[f[x],{x,0,500}, PlotRange->All]

The fact that f[x] = 0 is much clearer in these plots. This means that the line y = 0 is a horizontal asymptote for the right side of the graph. Notice that the function's graph crosses this asymptote many times. What matters is that for large values of x, f(x) gets arbitrarily close to 0.
Also note that since f[x] does not exist, this is an example of a function with a one-sided horizontal asymptote.