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##\displaystyle\lim_{x\rightarrow 1}\frac{(x+1)(x-1)}{x - 1}##

##\displaystyle\lim_{x\rightarrow 1}x+1##

##2##

But what justification do we have for cancelling the (x - 1) terms? When we cancel these terms, we are effectively dealing with a whole new function since the domain changes. Why is changing functions allowed in doing limits? How can we be absolutely certain that ##\lim_{x\rightarrow 1}x+1## leads to the correct answer to the original problem if ##x + 2## is different function, with a different domain, than ##\frac{x^2 -1}{x - 1}##?