Math CentralQuandaries & Queries


Question from Carina, a student:

Hi. My name's Carina and I'm currently a sophomore in high school.
I'm having a lot of difficulties in AP Calculus with continuity,
one-sided limits, and removable discontinuities. Basically, I have no
idea how to do them or even what they are. I read the lesson but I
still don't get it. Can someone put it in simpler terms so I can
understand how to complete my questions? Thank you!


A limit of a function at a value x can be thought of as the value that you would guess if you could not see what the value was at x but could see what it was nearby. The limit is "what its friends say about it"

So, for instance, the function f(x) = x/x has the value 1 everywhere except at x=0, where it is undefined. But by looking at values such as f(1) = 1, f(1/10) = 1, f(1/10000) = 1... we decide that if f(0) did have a value it should be 1.

If the limit exists at a point and equals the value of the function it's called "continuous". (Its friends' descriptions agree and are accurate.) The function f(x) = 2x is continuous at 1: looking at f(1.1) = 2.2, f(1.01) = 2.02, f(1.001) = 2.002... we conclude that f(1) ought to be 2; and it is.

If the limit doesn't exist (the friends can't agree), or it exists but isn't the real value, the function is discontinuous. The second case (the friends agree but are wrong, like in "Easy A") is called a "removable singularity".

A function has a one-sided limit from the left at a point if the values to the left give a consistent idea of what the value should be there - even if they are wrong, and even if the values on the right suggest something else. A limit form the right is defined similarly. (Like if all your school friends think one thing about you and all your other friends have a different idea?) An example is x/|x|, which is -1 for every negative number and 1 for every positive number. So if you ask the negative numbers you'd guess that f(0) should be -1; if you ask the positive numbers you'd guess 1; and in fact it's undefined.

Good Hunting!


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