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}]}]