First, set your xlim and ylim values:
xlim = 5;
ylim = Limit[(x - 5)/(x^2 - 25), x -> xlim];
Then, use Epilog to draw a dot:
Plot[(x - 5)/(x^2 - 25), {x, -2.752, 12.752},
Epilog -> {AbsolutePointSize[5], Red, Point[{xlim, ylim}]}]
Or to draw a circle:
Plot[(x - 5)/(x^2 - 25), {x, -2.752, 12.752}, Epilog -> {Red, Circle[{xlim, ylim}, Scaled[{.02/GoldenRatio, .02}]]}]
Or something more fun:
Plot[(x - 5)/(x^2 - 25), {x, -2.752, 12.752},
Epilog -> {Text[Style["\[FreakedSmiley]", 36], {xlim, ylim}]}]
