This trick for plotting the asymptotes only works when there is a jump discontinuity: the exclusion will show the jump.
When the function goes to +infinity from both sides (or the functions only exists on one side) there is no jump to show.
I am afraid we have to draw those asymptotes manually, with Exclusions for example.
Plot[1/x, {x, -1, 1}, Exclusions -> x == 0,
ExclusionsStyle -> Directive[Red, Thick]]
Plot[1/x^2, {x, -1, 1}, Exclusions -> x == 0,
ExclusionsStyle -> Directive[Red, Thick]]
Plot[1/x^2, {x, -1, 1},
Epilog -> {Directive[Red, Thick, Dashed],
InfiniteLine[{{0, 0}, {0, 1}}]}]
