[Edited to include the correction noted in the comment]
From what I can tell, no, you can't get that information directly.
But it should match the min and max values of the plotted data, so a work-around might be evaluate the axes children and find the maximum values.
Alternately, you could change the Limit Mode to "auto" (app.axes.XLimMode = "auto") and the Limit Method to "tight" (app.axes.XLimitMethod = "tight"). That should provide the same visual result as setting app.axes.XLim = [-Inf, Inf], but allows you to access the automatically calculated value with something like x = app.axes.XLim
