BQ #6: Unit U Concepts 1-8

1- What is a continuity? What is a discontinuity?
 A continuity is when the function is predictable, has no breaks, holes, or jumps, and can be drawn without lifting your pencil. A discontinuity is when the function DOES have holes, breaks, or jumps, and it isn't predictable. As well does a discontinuity have two types of families:
The first type of family is the Removable Discontinuities- This family only has the Point Discontinuity, which is pretty much a graphed out line with a hole in it (it can also be a hole with a point above it).
 Ex.

http://bfreshrize.files.wordpress.com/2012/01/unknown.jpeg?w=500

The second type of family is the Non-Removable Discontinuities- This family has three discontinuities. The first one is the Jump Discontinuity, which is similar to piecewise graphs however both parts cannot have closed points but can have them both open or one open and one closed.
Ex.

http://image.tutorvista.com/content/feed/u364/discontin.GIF

The second one is Oscillating Behavior, which is just a line graphed out that is extremely "wiggly" that it makes it almost impossible to find the height, limit, and value of it.
Ex.

http://webpages.charter.net/mwhitneyshhs/calculus/limits/limit-graph8.jpg

The last one is the Infinite Discontinuity, which is also known as unbounded behavior and it occurs were there is a vertical asymptote.
Ex.

http://www.math.brown.edu/utra/discontinuities%201.GIF

2- What is a limit? When does a limit exist? What is the difference between a limit and a value?
 A limit is the intended height of a function. The limit exists whenever you reach the same height from both the left and right (on a continuity). However it does not exist when you cannot reach the same height from both the left and right (on a discontinuity). The difference between the two is that a limit is the intended height while the value is the actual height of the function.
3- How do we evaluate limits numerically, graphically, and algebraically?
When evaluating limits numerically you just have to create a table to find the points closest to the value and vise versa to find the limit.
Ex.

http://media.showme.com/files/349881/pictures/thumbs/658760/last_thumb1359484583_200x150.jpg

To find limits graphically, we use our fingertips and follow the line to the point that we are looking for.
Ex.

http://www.mathnstuff.com/math/spoken/here/2class/420/c42d.gif

When finding limits algebraically, we use three methods: Direct Substitution, Dividing out/Factoring, or Rationalizing/Conjugate. However when finding limits algebraically we must always start with direct substitution to make sure that it equals 0/0 for we can do anymore, for if it equaled a real number to begin with all of the work done would have wasted time.
Ex. (Substitution)

Ex. (Factoring)

Ex. (Rationalizing)