Take f[x] = Sin[1/x]. Make several plots of f[x] to see what happens as
x -> 0. Does f[x] exist? Explain why or why not.

Answer:

Clear[f];
f[x_]=  Sin[1/x]

Here are the plots zooming in on what happens as x-> 0:

Plot[f[x],{x,-10,10},PlotRange->{-1,1}]

Plot[f[x],{x,-1,1},PlotRange->{-1,1}]

Plot[f[x],{x,-.1,.1},PlotRange->{-1,1}]

Plot[f[x],{x,-.001,.001},PlotRange->{-1,1}]

Certainly not nice behavior! As x gets closer and closer to 0, the values of f[x] still range all the way from -1 to +1. The graph does not seem to be settling down and approaching a single point as x->0. Since there is no trend in the y-values, we have no hope that f[x] exists.