Question about continuous function

A few questions about limits and continuity.

1)Am I correct that for a continuous function, the definition of a limit allows x=a since abs(x-a) =0 -> abs(F(x)-F(a))=0< e?

2) I have come across a function which I need to find the limit of at x=1. It requires $\displaystyle lim(x^2+1) , x=1$ when x^2+1 is not defined at x=1. Now I realise that for limits we are not interested what happens at x=1. It's clearly not continuous at x=1 so I seemingly cannot apply limit of F(x)=F(1). THe answer is obviously 2 but how do I justify that answer?

Thanks for clearing up these simple questions.