Melissa's Math Analysis Journey: BQ#6: Unit U

1. What is a continuity? What is a discontinuity?
A continuity is a continuous function that is predictable. It has no breaks, jumps, or holes. It can be drawn without lifting your pencil off the paper. In a continuous graph, the intended height of the graph is the actual height. The limit and the value are the same.

http://www.math10.com/en/algebra/functions/function-continuity/11.gif

A discontinuity is when a graph is not continuous and it has breaks jumps or holes. The intended height of the graph does not equal the actual height. Discontinuities are broken down into two families. These two families are Removable and Non-Removable Discontinuities. There is one type of discontinuity that is a Removable Discontinuity, and that would be a point discontinuity, or a hole. The three Non-Removable discontinuities are jump, oscillating behavior, and infinite. A jump discontinuity is when there is a jump between two points which was caused by different left/rights. An oscillating discontinuity is a discontinuity that can be described as "wiggly." An infinite discontinuity is caused by a vertical asymptote, and this results in unbounded behavior.

http://www.ops.org/high/north/Portals/0/ACADEMICS/StaffPages/holleyd/calculushtml01/calc1-4notes8.gif

Point Discontinuity

http://upload.wikimedia.org/wikipedia/commons/e/e6/Discontinuity_jump.eps.png

Jump Discontinuity

http://web.cs.du.edu/~rjudd/calculus/calc1/notes/dis6.png

Oscillating Behavior

http://dj1hlxw0wr920.cloudfront.net/userfiles/wyzfiles/44bad38c-431e-4382-8fe9-86303561b2a0.gif

Infinite Discontinuity

2. What is a limit? When does a limit exist? When does a limit does not exist? What is the difference between a limit and a value?

A limit is the intended height of a function. A limit exists when the left and right side meet. This is the reason on why a limit still exists at a point. Both sides intended to go there, so the L/R are the same number. As long as you reach the same height from the left and right, a limit exists. A limit does not exist when there are different L/R. When the limit does not exist, it is because of jumps, breaks, or holes. The value is thr actual height, while the limit is the intended height.

http://www.formyschoolstuff.com/school/math/glossary/images/JumpDiscontinuity.gif
Limit does not exist because of a jump discontinuity; different L/R

http://web.cs.du.edu/~rjudd/calculus/calc1/notes/dis4.png
Limit does not exist because of unbounded behavior

3. How do we evaluate limits numerically, graphically, and algebraically?
When evaluating a limit, there are 3 possible ways to do this: numerically, graphically, and algebraically. Evaluating a limit numerically means on a table. To do this, we must place the number that x is approaching in the center of the graph, then we must add/subtract .1 from this number. We can then use a calculator to find the f(x) values at these x-values. The table helps us determine the x and f(x), the intended and actual height of the function. An example can be seen below.

http://i1.ytimg.com/vi/uTyYPfwVEGo/maxresdefault.jpg

To evaluate a limit graphically, we use a graph. When you have a picture of the graph, it can be done by using your fingers. To do this, we must place our fingers to the left and right of where we want to evaluate the limit.We do this to see if the right and left side meet. If our fingers meet, then there is a limit. The limit exists at the point our fingers meet. If our fingers do not meet, then the limit does not exist. We can also use a calculator to evaluate a limit graphically. To do this, we must simply plug in the function to the calculator.

http://s3.amazonaws.com/KA-youtube-converted/XIsPC-f2e2c.mp4/XIsPC-f2e2c.png

To evaluate a limit algebraically, we must first use something called direct substitution. In this method, we must substitute the number it is approaching into te equation. If the answer ends up being 0/0(indeterminate form), only then can we use the dividing out/factoring method and the rationalizing/conjugate method. The dividing out/factoring method involves factoring out both the numerator and denominator in order to cancel out some terms. When the expression is simplified, you can then plug in the number into what you have left. The rationalizing/conjugate method is used when all of the other methods fail. To do this, we must multiply the function by the numerator or denominator, depending which one has the radical. Then, we must then FOIL, but we must leave the non-conjugate denominator factored. We must then use direct substitution on the simplified expression.