BQ #6
1. Q: What is a continuity? what is a discontinuity?
A continuity is a continuous function that is predictable and has no breaks, holes, or jumps. It also means that it can be drawn without your pen/pencil having to be lifted off the paper. A function is continuous if the intended height of a function and the actual height of a function are the same.
Ex.
(This is a continuous function that has no breaks, holes, or jumps, it can be drawn w/o your pencil leaving the paper)
A discontinuity happens when the intended height of a function and the actual height of a function are different. This can result in one of three different discontinuities, either jump, infinite, or oscillating. A jump discontinuity is when there is a jump between two points that was caused between different left/rights. Secondly, is the oscillating discontinuity, which we mathematicians taught by Mrs. Kirch describe with our astounding vocabulary as, wiggly. Lastly, another discontinuity is an infinite discontinuity which is caused by a vertical asymptote which results in unbounded behavior.
(This is an example of a jump discontinuity, as you can see there is a jump between 2 points and comes in from one left but it leaves in a different right)
(This is an example of an oscillating discontinuity which when looked at, is quite wiggly indeed)
(This is an example of an infinite discontinuity which as you can see has a vertical asymptote that results in it having unbounded behavior)
2. Q: What is a limit? When does a limit exist? When does a limit not exist? What is the difference between a limit and value?
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. A limit does not exist if the left and right hand limits are not equal, this can usually be caused by breaks, jumps, or holes in the graph and function. A limit is the intended height of a function while a value is the actual height of a function.
(As you can see, even though we come in from different sides on the left and the right we end up at the same spot in the middle, this is an example of an existing limit.)
(This is an example of a limit NOT existing, as you can see we start from different sides but unlike the previous picture, we don't end up at the same place in the middle, instead there is a "jump discontinuity")
3. How do we evaluate limits numerically, graphically, and algebraically?
While multiple ways to evaluate limits, we will be focusing on three specific ways of evaluating them, those three being numerically, graphically, and algebraically. Numerically, we evaluate limits using tables which help us to determine the "x" and the "f(x)", which are the intended and actual height of a function. An example can be shown in the below picture.
(This picture shows a limit being evaluated numerically)
Another way of evaluating limits is by doing them is graphically. There are many ways to do it graphically but there are two main ways, those being having a picture of a graph and using your fingers, the other way is using a graphing calculator by plugging in the equation of the function. The following picture shows doing it by plugging in the equation of the function into your graphing calculator.
(As you can see, even though we come in from different sides on the left and the right we end up at the same spot in the middle, this is an example of an existing limit.)
(This is an example of a limit NOT existing, as you can see we start from different sides but unlike the previous picture, we don't end up at the same place in the middle, instead there is a "jump discontinuity")
3. How do we evaluate limits numerically, graphically, and algebraically?
While multiple ways to evaluate limits, we will be focusing on three specific ways of evaluating them, those three being numerically, graphically, and algebraically. Numerically, we evaluate limits using tables which help us to determine the "x" and the "f(x)", which are the intended and actual height of a function. An example can be shown in the below picture.
(This picture shows a limit being evaluated numerically)
Another way of evaluating limits is by doing them is graphically. There are many ways to do it graphically but there are two main ways, those being having a picture of a graph and using your fingers, the other way is using a graphing calculator by plugging in the equation of the function. The following picture shows doing it by plugging in the equation of the function into your graphing calculator.
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| (This shows a picture of solving it graphically by plugging in the equation into your graphing calculator.) The last way of evaluating the limit is by doing it algebraically. We do this by plugging the limit that is given into the function itself, substituting the x with the limit. All you would need is the limit and the function to evaluate the limit this way. The following picture shows an example of this being done in a step by step process.  (This picture shows a limit being evaluated algebraically) Sources
Picture 1- http://www.mathsisfun.com/calculus/continuity.html
Picture 2- commons.wikimedia.org/wiki/File:Jump_discontinuity_cadlag.svg
Picture 3- http://web.cs.du.edu/~rjudd/calculus/calc1/notes/discontinuities/
Picture 4- http://www.wyzant.com/resources/lessons/math/calculus/limits/continuity
Picture 5- http://www.wyzant.com/resources/lessons/math/calculus/limitsPicture 6-http://curvebank.calstatela.edu/limit/limit.htm Picture 7- https://www.youtube.com/watch?v=QQEeZB6m6e0 Picture 8- https://www.youtube.com/watch?v=eGEW67LdpRE Picture 9- https://bccalculus.wikispaces.com/Algebraic+Methods+for+Evaluating+Limits |
