1) jump - A jump exists in a graph when the left and right sides of the limit do not approach the same thing. It is called a jump, because the function jumps from one point to another (limit to limit) and therefore it is discontinuous. This is very prevalent in piecewise functions.
Example:
f(x) =
{x^2 for x is less than 1
{0 for x=1
{2 - (x-1)^2 for x is greater than 1
If you plug in, you will find that there are two different values for 1, 1 and 2. Therefore it is a jump.
2) removable - Removables are mostly seen in factorable functions. It is called removable because you can essentially remove it and the limit could exist continuously after factorization.
Example:
the limit as x approaches 3 of the function
(x^2 - 10x + 21) / (x^2 + x -12)
As all should know, if you plug 3 directly in, you will get an undefined answer. Therefore, you must factor to be able to get a finite limit.
So, that factors out to...
((x-7)(x-3))/((x+4)(x-3))
You can cross out (x-3) on the top and bottom (canceling), therefore you set what you cancel equal to 0 and that is the removable. So x=3 is removable.
3) vertical asymptote - You find this almost the same as a removable. When there is nothing left to factor, you set the bottom equal to 0.
Example:
the limit as x approaches 3 of the function
(x-5)/(x-3)
set x-3=0
x=3 is your vertical asymptote
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