Limits & Functions
1. What is continuity? What is discontinuity?
(http://tutorial.math.lamar.edu/Classes/CalcI/Continuity.aspx)A: A continuous function is predictable. It has no breaks in the graph, no holes, and no jumps. A continuous function can be drawn with a single, unbroken pencil stroke. A continuous function makes a good bridge. You can't fall thru or fall off. A function is continuous if lim_x->c f(x) = f(c). A discontinuous function is the opposite of what a continuous function is. It has breaks in the graph, holes, and jumps. A discontinuous function cannot be drawn with a single, unbroken pencil stroke. There are 3 types of discontinuities [in two families]: Removable discontinuities: which is know as the point discontinuity. Then there's the Non-Removable discontinuities: Jump (different L/R , Oscillating Behavior (wiggly), and Infinite discontinuity due to unbounded behavior.
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: A limit is the intended height of a function. A limit exists as long as you reach the same height from both the left and the right (example: As long as you and your friend both arrive at the same restaurant when coming from different directions.) 
(http://www.millersville.edu/~bikenaga/calculus/limlr/limlr.html) In order for a limit to exist, both the right hand limit and left hand limit must be the same. If a graph does break at a given x-value, a limit exists there. A limit can exist if your ultimate destination is a hole in the graph! (i.e. You and your friend both drive to the restaurant, but the restaurant isn't there. If you both arrived at the same "parking lot", the limit still exists!") The limits also don't exist when we're comparing left and right behavior, unbounded behavior, and oscillating behavior. The difference between a limit and a value is that a limit is the intended height, while a value is the actual height.
(http://www.millersville.edu/~bikenaga/calculus/limlr/limlr.html) In order for a limit to exist, both the right hand limit and left hand limit must be the same. If a graph does break at a given x-value, a limit exists there. A limit can exist if your ultimate destination is a hole in the graph! (i.e. You and your friend both drive to the restaurant, but the restaurant isn't there. If you both arrived at the same "parking lot", the limit still exists!") The limits also don't exist when we're comparing left and right behavior, unbounded behavior, and oscillating behavior. The difference between a limit and a value is that a limit is the intended height, while a value is the actual height.
3. How do we evaluate limits numerically, graphically, and algebraically?
a: In order to evaluate we must know what the term evaluating means. Evaluate means you must give a numerical answer. There are three techniques for evaluating: substitution, factoring, and the conjugate method. To evaluate limits graphically we can go to our calculator and hit the Graph button, 2nd -> hit trace, and lastly trace to the value that you're looking for. Then there's also the method of direct substitution which means you take the number the limit is approaching and plug it in anytime you see "x" and this later converts to evaluating the limits algebraically. Numerically, is a chart that helps us and supports us to find the answer that we're looking for since we're a using a chart.
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