For this limit to exist requires that

lim(x->1+) f(x) = lim(x->1-) f(x)

and both be finite (lim(x->1+) means the limit as x goes to 1 from above,

and lim(x->1-) means the limit as x goes to 1 from below.]

Now:

lim(x->1+) f(x) = lim(x->1+) x+1 = 2

and

lim(x->1-) f(x) = lim(x->1-) -x^2 = -1.

Hence:

lim(x->1+) f(x) != lim(x->1-) f(x)

and so the limit does not exist.

RonL