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}}]}]