1. What is continuity?
Continuity is a way to describe a function that is predictable. It has no breaks, jumps, or holes. This kind of function can be drawn on a graph without lifting your pencil. In addition, the value is the same as the limit.
What is discontinuity?
Discontinuity is a way to describe a function that is not predictable. There are two families: removable and non-removable discontinuities.
A point discontinuity is a removable discontinuity. Here the limit exists wherever the open circle is. This looks like a hole on the graph. Sometimes the value and the limit might not be at the same point. In this case, it is as if you and your friend plan to go to the mall and show up there, but it burned down.
In a non-removable discontinuity, the limit does not exist (NOT "there is no limit"). This includes a jump discontinuity where the graph is different coming from left and right. In this case, it is as if you and your friend plan to meet at a restaurant, but you two show up at different ones. Oscillating behavior has no limit and the value is undefined because it is simply oscillating and appears wiggly on a graph. In an infinite discontinuity (unbounded behavior) has vertical asymptote. This means it increases or decreases without bound toward infinity or negative infinity.
2. What is a limit?
A limit is the intended height of a function. We read a limit as: "The limit as x approaches 'a number' of f(x) is equal to 'L.'" You can have an infinite amount of limits on a graph.
When does a limit exist?
The limit exists in continuous functions. This is when the intended is the actual height also. However, we must understand that the function is not the value. The three times a limit does not exist are when we have non-removable discontinuities. When it 'DNE,' we must also right the reasons. If it was a jump discontinuity, the graph is different left and right. If it was an infinite discontinuity, it has unbounded behavior. If it is oscillating, the graph is simply oscillating and we cannot find its value either which means it is undefined.
What is the difference between a limit and a value?
The limit is the intended height of the graph. Often this can be seen with two open circles or an open and closed circle. The value is the actual height. This is shown with a closed circle.
3. How do we evaluate limits numerically, graphically, and algebraically?
We can evaluate limits numerically which is on a table. The tables I have drawn in the pictures show how we can get really really really really close to a number without actually touching it.
Evaluating limits on a graph involves an actual graph. You put two fingers on the left and right side of where you want to evaluate a limit. If your fingers do not touch, the limit DNE.
Algebraically, you should first try the direct substitution method. We can get a number, 0 over a number which is 0, a number over 0 which means the limit DNE, or 0/0 which is the indeterminate form. If you get 0/0, use another method because it is "not yet determined" and we need to keep on working to find the answer. Whatever number the x approaches, you plug that into any variable of the equation given. You can also try the rationalizing/conjugate method. Here you can use either the conjugate of the denominator or the numerator. You need to rearrange the equation given though by multiplying the top and bottom by the conjugate of either the numerator or denominator. Whichever conjugate you used, simplify it by FOILing. The non-conjugate will be left alone, so you can cancel things out. Remember, "...if something cancels, then your graph will have a HOLE!"
