1. What is continuity? What is discontinuity?
Continuity means the graph is predictable; it has no breaks, holes, or jumps. It can be drawn without lifting the pencil off the paper.
Discontinuity is when the graph can contain breaks, holes, and jumps. It's separated into two families: Removable Discontinuities and Non-Removable Discontinuities. Point discontinuity is a removable discontinuity, meaning there's a hole in the graph. Non-Removable Discontinuities has Jump Discontinuity, Oscillating Behavior, and Infinite Discontinuity. Jump discontinuity means they have different left and right. You must remember that the jump can have two open circles, one close/open, but they must never be two closed circles. Oscillating behavior means the graph is very wiggly, or unpredictable. The limit DNE because you can't really pinpoint the graph's next destination. Infinite discontinuity is when there's a vertical asymptote leading to unbounded behavior. Unbounded behavior means it increases or decreases without bond. This is because the graph is either going up infinitely or down infinitely.
2. What's a limit? When does a limit exist? When does a limit DNE? What's the difference between limit and value?
A limit is the INTENDED height of a function; whereas a value is the ACTUAL height of a function. A limit exists when the intended and actual height are the same. A limit does not exist when the left and right are different, unbounded behavior, and oscillating behavior.
3. How do we evaluate limits numerically, graphically, and algebraically?
Numerically means you use the table. As the table goes towards the center, it's getting closer to the limit, or y-value. It also helps us find if the limit can be reached or not. Graphically is when we plug the function onto our calculator to graph and then trace to the value you are looking for. To evaluate the limit, you put your finger on a spot to the right and left; where your fingers meet is where the limit is. If your fingers does not meet, then it does not exist. For algebraically, there are three methods: Substitution, Factoring, and Conjugate. Substitution is simply just plugging in the number into the function. You always want to try this method first before doing any other method. If you get 0/0, that means it's an indeterminate form and you have to use another method. The factoring method is when you factor the numerator and denominator and cancel like terms to remove the zero. Then with the simplified expression, use substitution to find your answer. The rationalizing/conjugate method is when you multiply the radical with its conjugate. Remember to not multiply the non-conjugate because something would hopefully cancel. Once something does cancel, you use substitution to get your answer.