-In a nutshell, a continuity is basically a graph which is completely predictable. These graphs are the types of graphs we have been dealing through almost all of algebra. A continuous graph has no points, holes, jumps, and always ends "landing"in the predicted location. On the other hand, a discontinuity is a graph in which you do NOT always end up in the location that is predicted. This can mean that there is either a point, a hole, or a jump. There are two kinds of discontinuities: removable and non-removable. If you wish to know what this means, then by all means, keep reading.
Removable:
The Point discontinuity shown below is known as a hole, is characterized by the hole that is has in he middle, and is the only Removable discontinuity. This is because the graph is technically continuous. This is because you go from negative infinity to positive infinity.
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| http://bfreshrize.files.wordpress.com/2012/01/unknown.jpeg?w=500 |
Non-Removable
The Jump discontinuity shown below is characterized by its similarity to the hole discontinuity. However, upon close inspection, we can see that it different because it resemble a cliff; thus the name "jump discontinuity".
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| http://upload.wikimedia.org/wikipedia/commons/e/e6/Discontinuity_jump.eps.png |
The Oscillating Behavior discontinuity shown below is characterized by its wiggly aspect.
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| http://webpages.charter.net/mwhitneyshhs/calculus/limits/limit-graph8.jpg |
The infinite discontinuity is practically what we know as a vertical asymptote. This discontinuity is known as unbounded behavior and exists when there is a vertical asymptote.
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| http://web.cs.du.edu/~rjudd/calculus/calc1/notes/dis3.png |
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 limit is the intended location that you planned to reach. The limit exists whenever you reach the same location when approaching from both Left and Right. THE LIMIT DOES NOT EXIST!!!!!!!!!1 whenever you do not reach the same location. For example, if Mrs. Kirch and Mrs. Lund intend on meeting in front of the ASB window and they end up in front of the window, then the limit exists. However, if Mrs. Kirch somehow ends up in front of the attendance window while Mrs. Lund waits in front the ASB window, the limit does not exist. This means that in a jump discontinuity, the limit doesn't exist because you reach different locations when going from left and right. In infinite discontinuity, you never reach the intended location because the vertical asymptote is there. In the oscillating behavior, we do not have a limit because you graph never approaches a specific point on the graph. The difference between a limit and a value and the limit is that the limit is the INTENDED location while the value is the ACTUAL location you reach.
3. How do we evaluate limits numerically, graphically, and algebraically?
In order to evaluate the limit numerically, you start out with the intended "height" and fill in the rest of the table. Usually, we have the intended height in the middle, and then three blank boxes that signify where the graph crosses (x axis). In order to solve graphically, we look at the limits we are asked to find. Then, we look at the graph and find the location we are asked to find, then simply use our knowledge of discontinuities to solve. For example, we know that in a jump discontinuity the limit doesn't exist because you approach two heights and in the point discontinuities, the limit would be the hole. In order to solve algebraically, we ALWAYS try directly substituting the given limit. If that doesn't work out, then we factor both the numerator and the denominator (if both are possible) and then just go wild and (in an adrenaline pumping fashion) eliminate those common terms. When we have fully reduced, we plug in the limit to find the answer. Another way is to us the Rationalizing and Conjugate method. In order to do this, we simply multiply the conjugate of either the top of the bottom (it depends on where the radical is). Then, we multiply the outer and inner parts of the numerator. We NEVER EVER EVER EVER foil out the denominator. This will just give you a problem that is a godzillion times more difficult. Anyways, we simply reduce and then plug in the limit. Keep in mind that we always try to directly substitute.
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