The function is defined: why?

I am looking at the graph of function f(x)=(x-1)/(x^2-1) produced by the program: "Graph" created by Ivan Johanson.

Why is this rational function defined at x = 1 as the limiting value 0.5?

Is this a convention for it to be defined as the limit? Or am I crazy? (Nerd)

Well, you are only crazy if you expect a computer to be as good as your brain! What is graphed there is NOT the function you give. What is true is that

which is not defined at x= 1 and so its graph should have no point on the vertical line at x= 1.

However, for all x except x= 1,

which is defined at x= 1. The computer, calculating values at a finite number of values of x, "misses" the problem at x= 1 and just graphs y= 1/(x+1).
The graph is wrong.

Thanks for clarifying.