1. What is a continuity? What is a discontinuity?
A continuity is a predictable graph in which you can draw without lifting your pencil and does not have jumps, breaks, or holes. To see an example of a continuous graph look at the first image below. A discontinuity on the other hand consists of non removable and removable. A non removable is non removable for the fact that there are different left and right heights on both sides of the graph, the graph is oscillating meaning it is extremely wiggly, or there is an unbounded behavior meaning that there is a vertical asymptote. At vertical asymptote keeps going and going closer and closer to the graph. When saying there are different left and rights, it simply means that if you put your hand on the left side and the right side of the graph and slide them together, they do not meet. A removable discontinuity is a point discontinuity meaning that there is a hole in the graph, as seen in the second image below.
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| http://www.analyzemath.com/calculus/continuity/continuous_1.gif |
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| https://encrypted-tbn0.gstatic.com/images?q=tbn:ANd9GcTfGXTPwPhR-q7-0XVjAsFKgDYLVYvmiIoGlO2I_UFAHoiBrlgl |
2. What is a limit? What is a discontinuity?
A limit is the intended height of a function, meaning its where two graphs are to meet up with each other. It is not able to exist when there is an unbounded behavior meaning there is a vertical asymptote, it is oscillating, and there are different left and right heights. An example of each is shown below.
Unbounded behavior
http://shell.cas.usf.edu/~ssuen/2233.2013s/c/chapter1.diagrams/1.6.18.png
Different left and right
http://image.tutorvista.com/content/feed/u364/discontin.GIF
Oscillating
http://webpages.charter.net/mwhitneyshhs/calculus/limits/limit-graph8.jpg
3. How do we evaluate limits numerically, graphically, and algebraically?
Numerically means to set up a table. You are to have the limit in the middle and three boxes on each side. On far right side box you add 1/10 and the far left you subtract 1/10. From each sides you are to see clearly that the numbers keep getting a little closer and closer. To see an example look below. To evaluate a limit graphically you simply put a hand on the left and right side of the graph and slide them together. To evaluate it algebraically you may use substitution, which is simply plugging in the x value and solving. nalizing/conjugate method which is multiplying the conjugate and canceling. A conjugate is taking the radical expression and changing its sign.
Works Cited
http://www.analyzemath.com/calculus/continuity/continuous_1.gif
https://encrypted-tbn0.gstatic.com/images?q=tbn:ANd9GcTfGXTPwPhR-q7-0XVjAsFKgDYLVYvmiIoGlO2I_UFAHoiBrlgl
http://shell.cas.usf.edu/~ssuen/2233.2013s/c/chapter1.diagrams/1.6.18.png
http://image.tutorvista.com/content/feed/u364/discontin.GIF
http://webpages.charter.net/mwhitneyshhs/calculus/limits/limit-graph8.jpg
