BQ #6- Unit U

What is continuity? What is discontinuity?

A continuity is basically a predictable graph  or a continuous graph.  A continuity is also described as a graph that "has no breaks, no holes, and no jumps". Another way you can identify a continuity is by drawing it without lifting  your pencil from the paper and makes a goof bridge. We are more familiar with continuities as we dealt with them in parabolas, linear graphs, etc. Meanwhile, discontinuities are the opposite of continuities since they do have holes, jumps, and breaks. Discontinuity graphs are categorized into two different "families". The two families are removable discontinuities and non- removable discontinuities. In removable discontinuities, the one example would be point discontinuities, or a hole. Non-removable discontinuities have three different examples: jump discontinuities, oscillating behavior, and infinite discontinuity. Jump discontinuities are graphs that have an open and closed circle, but do not connect. An oscillating behavior is a graph that looks sine curve, but is very wiggly. An infinite discontinuity is also known as unbounded behavior because it occurs where there is a vertical asymptote.

Removable Discontinuity:

Point Discontinuities

http://dj1hlxw0wr920.cloudfront.net/userfiles/wyzfiles/4a69dec7-03e0-492f-ac16-4dcd555579c9.gif

Non-Removable Discontinuities

Jump Discontinuity

http://upload.wikimedia.org/wikipedia/commons/thumb/9/92/Jump_discontinuity_cadlag.svg/264px-Jump_discontinuity_cadlag.svg.png

Oscillating Behavior

https://encrypted-tbn3.gstatic.com/images?q=tbn:ANd9GcRGKnu-whU8KA1HjLrbu6pgV9JW_727lNXa-Eu-3sTzcp_zGqukDg

Infinite Behavior

https://encrypted-tbn2.gstatic.com/images?q=tbn:ANd9GcRp9ghtAEQLeV_IInaJpdPzTWbn-xansC05qQdrQ_6jz_yURt2TQQ

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 a function". As for the values, it is the actual height of a graph. Keep in mind that a graph is continuous when the limit and the value are the same and we have no discontinuity. So now let's go back, where does a limit exist/ not exists?
A limit exists in any point within a continuous function and also exists in a hole because the graph still intended to reach a certain height. As for the value, the random point above would be the actual value since it is the actual height of the function (dark circle).

Analogy of why the limit still exists in a hole:

Two people are driving to a restaurant from two different routes. Once they get there, they noticed that the restaurant burned down. Despite the fact that they did not go inside the restaurant, the two different people still reached the intended destination- the restaurant.

http://dj1hlxw0wr920.cloudfront.net/userfiles/wyzfi

However, a limit will fail to exists in non-removable discontinuities, such as jump, oscillating, and infinite behaviors. The reason why a limit would not exist in a jump discontinuity is because there are different left and right limits. Also, the value and the jump discontinuity in jump discontinuities are different. When looking for the values, the value of a jump discontinuity would be the dark or closed circle. A limit will not exist in an oscillating graph because there are too many values within the "wiggly" graph that it is difficult to pinpoint one single value or limit. The last discontinuity in which a limit does not exist would be in infinite behavior graphs. This usually happens when we have a vertical asymptote, which would lead into unbounded behavior graphs. We will forever more approach infinity, which means we cannot find a specific numerical value.

How do we evaluate limits numerically, graphically, and algebraically? 
One of the ways to evaluate limits would be by using a table. We take the limit and add or subtract (1/10) to it.(three for the left and three for the right). Those two values would be the end points of the table and those f(x) values would approach the limit. 
The other way we can evaluate limits would be graphically. When we find values to our table, we can simply plug the function into the "y=" sceen and trace them. There are times when the limits are graphed and in those cases, it would be easier to trace the graph with our own fingers. If out fingers meet then the limit does exist, but it is doesn't then the limit does not exist. 
Lastly, we can also find the limits algebraically. We can evaluate algebraically with direct substitution, factoring out, and finding the conjugate. We always want to start solving with direct substitution, which is basically substituting the x value that we are approaching into the function. You can will be done with the problem if you get a # (or fraction), 0/#, #/0, or 0/0 (indeterminate). If you do get indeterminate, then use the next methods. When we factor out the function we should be able to cancel something out (removable) that would result in a simply number which we can directly substitute back into the function. The other method would be rationalizing/ conjugate and it is usually used when we have a radical and cannot factor out anything. We follow by multiplying by the conjugate of the portion that has a radical. We foil the conjugate and leave the non-conjugates alone. Again, we should be able to substitute once we canceled something else.

References

http://dj1hlxw0wr920.cloudfront.net/userfiles/wyzfi

https://encrypted-tbn2.gstatic.com/images?q=tbn:ANd9GcRp9ghtAEQLeV_IInaJpdPzTWbn-xansC05qQdrQ_6jz_yURt2TQQ

https://encrypted-tbn3.gstatic.com/images?q=tbn:ANd9GcRGKnu-whU8KA1HjLrbu6pgV9JW_727lNXa-Eu-3sTzcp_zGqukDg
http://upload.wikimedia.org/wikipedia/commons/thumb/9/92/Jump_discontinuity_cadlag.svg/264px-Jump_discontinuity_cadlag.svg.png
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