I'm looking for a way to make a plot similar to the one I've sketched. 
So far all I've found is this discussion, which mostly uses Grid. I want the second part of the plot to have a smaller interval, as shown, but when I try their technique, I get this plot:
Grid@{{Plot[Sin[2 x], {x, 0, 4}, PlotRange -> {-1.1, 1.1}, AxesOrigin -> {0, 0}], Style[Rotate["/", 0 Degree], 20], Plot[Cos[2 x], {x, 6, 8}, PlotRange -> {-1.1, 1.1}, Axes -> {True, False}]}} Notice how the scaling on the second plot changes. I'd like to keep the scaling the same between the first and second plot.

