Comments by "MC116" (@angelmendez-rivera351) on "Find the limit of (x - 1)/(x^2(x + 2)) as x approaches negative 2 from the right" video.

It is harmful advice to tell students that the correct way to evaluate a limit is to start by "plugging in" into the function. This is incorrect, and it fails whenever the function is discontinuous, which you cannot know unless you already know the limit. Below, I present the correct method to do this exercise. Let f(x) = x – 1, and let g(x) = x^2·(x + 2). lim f(x) (x —> –2, x > –2) = –3, and lim g(x) (x —> –2, x > –2) = 0. Therefore, lim f(x)/g(x) (x —> –2, x > –2) = lim (x – 1)/(x^2·(x + 2)) (x —> –2, x > –2) does not exist. However, you can say that as x > –2, x —> –2, f(x)/g(x) —> –∞. You can say this, because g(x)/f(x) < 0 as x > –2, x —> –2.
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@johnflorio3576 In that special case in particular, nothing goes wrong, but teaching this is still harmful in general, especially when there is a correct way to teach it. I do not understand why people online like to encourage this thing where "if the method produces the right answer, then it does not matter if it is wrong." No, it absolutely does matter. Mathematics is more about how you get to the answers than it is about the answers themselves.
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@johnflorio3576 Actually, I was wrong, and you are wrong too. It actually is incorrect to "plug in" to see if the numerator and the denominator are 0 if either of them is not continuous. My criticism still stands, and it is very valid criticism, because, given that this misinformation is taught at a global scale, as a tutor and teacher, I have encountered too many students who make mistakes based on the fact that they do not know the correct method for doing this. So, yes, it is harmful.
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