1. What is continuity? What is discontinuity?
Continuity is when we are able to draw the graph without lifting the pencil from the paper. So that means the graph cannot have any jumps, holes, or break. It is also predictable. This means that we now it is approaching infinity on both sides, either positive or negative. Now what is discontinuity? Look above, and picture the exact opposite of what I just explained. Discontinuities can be draw by lifting the pencil. So there are jumps, holes, and breaks. It is also unpredictable, meaning we do not know where it is headed from the left and from the right at any given point. Now, there are four discontinuities. There are two families of discontinuity: removable and no-removable. Removable contains point discontinuity. Non-removable contains point discontinuity, oscillating behavior, and infinite discontinuity.
2. What is a limit? When does a limit exist? When does a limit not exist? What is the difference between a limit and a value?
A limit is the intended height of the graph. That means the limit can be a hole on the graph, because it was in fact intended to go to that point on the graph. The limit exists only at removable discontinuities, which is specifically point discontinuity. It exists here only because the intended height is specific from both the left and the right of the graph. However, the limit does not exist at non-removable discontinuities, which are point discontinuities, oscillating behavior, and infinite discontinuities. It doesn't exist at point discontinuities because there are two separate intended heights coming from both the left and the right of the graph. It doesn't exist at oscillating behavior because the graph is so wiggly, that there is no specific point on the graph and in general, cannot make up it's mind on where it wants to go. Lastly, it doesn't exist at infinite discontinuity because it occurs when there is a vertical asymptote, which leads to unbounded behavior which means two separate directions from both the right and the left. The limit is the intended height however the value is the actual height, which only includes shaded in holes and lines on the graph, not open circles.
3. How do we evaluate limits numerically, graphically, and algebraically.
We evaluate limits numerically by using tables. We put numbers that are approaching a certain number from the left and the right. That way we can see what is the intended height. For graphically we observe the graph and use onr finger on each side and see if they approach at the same point. If its a hole but they both meet then its a limit. If there are two seperate points, then there is no limit. For algebraically, there are three methods. The first method you should alwats try first is substitution. Thats when we directly substitute x and solce to find the answer. If the answer is 0/0 we use factoring to see if we can cross out any equation which means there will also be a hole. After factoring and crossing out, we then substitute the x and solve. If we cant factor though, we use conjugate if there are any Radicals to find anything that can cancel. And that's how we solve (:
