![]() ![]() If a function can be factored, and the factoring removes a discontinuity, it seems more logical to me to state that the original function had a false or phantom discontinuity. I am comfortable performing such manipulations, as it makes inherent sense to me that the limits are equivalent but I wouldn’t know how to mathematically state or prove that such manipulations are correct. I believe mathematicians refer to such a discontinuity as a ‘removable discontinuity’. where a factored equation has removed a discontinuity which was present in the original unfactored equation, and that the limits are considered equivalent. ![]() I want to better understand this distinction as Sal has identified this situation several times in multiple videos, i.e. For I learned from the video on Epsilon-Delta proofs that a limit can still exists at 'a' despite a function being undefined at 'a'. Is there a mathematical theorem or proof that states the limit of a factored equation is equivalent to the limit of the respective unfactored equation as long as the factored equation is continuous at the point for which the limit is taken? Is this distinction covered by the Epsilon-Delta proof of limits. Sal goes on to say that the limits of f(x) and g(x) are equivalent at 'a'. And that f(x) is defined and is continuous at ‘a’. I'm assuming this means that g(x) is defined for all real numbers except 'a', and therefore is not continuous at 'a'. In the video, Sal states that f(x) and g(x) are equivalent except at ‘a’, and that f(x) is continuous at ‘a’. ![]() I’m assuming that g(x) is the original unfactored equation and that f(x) is the resulting factored equation. In his explanation he speaks of two functions f(x) and g(x). 8:20 in the video, Sal begins to explain that it is important to understand that the factored equation is not the same as the original unfactored equation. ![]()
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