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Lim x->1 [(x^2 - 1)/(x-1)]

Limx->1 [(x-1)(x+1)/(x-1)]

Lim x->1 [(x+1)] = 2 (when x=1 is substituted in)

I understand that they can have the same limits despite not both being defined at x=1, however, what I don't get is why the original f(x) isn't defined, but the second one is. How can two equivalent functions not be defined at the same points? Can't all functions be simplified by factorising without jeapordizing where they are actually defined? This makes no sense to me...