Is there a way for the "Plot[]" function to show singularities?

Questions
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1

For example, the equation: ((E^t)*(t-1)+1)/t^2 is well behaved and everywhere defined EXCEPT (of course) at t=0. However, since t–>0 from either negative or positive directions results in y–>0, the curve is everywhere smooth. When using Plot[] on such a curve, I’d like to indicate that at that point (or any such point) there is a singularity. In texts I often see such things indicated by an open ‘dot’ at singularities (a tiny “o”; there is probably a correct typesetting term which I do not know). Is there a switch or command in Mathematica to force such indications during Plot[]? Thanks you for considering!

2

Plot the reciprocal of your function. The singularities will appear as zeros.