I am curious about the process of evaluating a limit. Firstly, I know that if a function ##f(x)## is continuous then one can usually just plug in the the number that ##x## is approaching in the limit, since criteria for a continuous function is that ##\lim_{x \to a}f(x) = f(a)##. However, what if we have the function ##f(x) = \frac{(x-1)^2}{(x+1)}##, then if we try to evaluate the limit ##\lim_{x \to -1}\frac{(x-1)^2}{(x+1)}##, we do so by cancelling the linear factors to get ##f(x) = x - 1##, which then gives ##\lim_{x \to -1}(x - 1) = -2##. But what is the justification that we're allowed to cancel out the factors when evaluating the limit? This produces an entirely new function, since ##f(x) = x - 1## is obviously different than ##f(x) = \frac{(x-1)^2}{(x+1)}##. Why are we allowed to manipulate one function into another in this way when evaluating limits? What guarantees that ##\lim_{x \to -1}(x - 1) = \lim_{x \to -1}\frac{(x-1)^2}{(x+1)}##?(adsbygoogle = window.adsbygoogle || []).push({});

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# Justification for evaluation of limits?

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