Unit U Big Questions

What is continuity?

In mathematical terms, continuity occurs when a graph has no breaks, holes, or jumps. Basically, the graph must be continuous. We can identify a continuous graph, or continuous function, when we see that is makes a good bridge. A function is considered continuous if the limit as "x" approaches "c" of f(x) equals f(c). [f(c) is the height the function reaches at that point of x=c]
The picture below shows a continuous function as well as a bridge, to show and complete the analogy: a continuous graph makes a good bridge. *Notice there are no breaks, holes, or jumps.

What is discontinuity?

A graph is considered discontinuous when there are breaks, holes, vertical asymptotes, "wiggly" lines, and/or jumps.
When a graph has a hole, this is called point discontinuity. Point discontinuity is under the category of REMOVABLE discontinuities because a hole can easily be "fixed" by filling it in to make a real point.

When a graph has a jump, this is called jump discontinuity. A jump is when the graph breaks at a certain x value with two different y-value (different left/right) see below:

When a graph has a vertical asymptote, this is called infinite discontinuity.

When a graph is "wiggly", this is called oscillating behavior.

Important Note: NONREMOVABLE discontinuites are: jump discontinuity, oscillating behavior, and infinite discontinuity.

For further clarification, watch this video to see the difference between continuity and discontinuity :)

What is a limit? When does it exist? When does it 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 limit on the right and on the left are the same. Note: A limit can still exist if your final destination is a hole in the graph. A limit does not exist when the graph has a break, such as when it exhibits oscillating behavior, unbounded behavior, or a jump discontinuity. A value is a number that can always be defined on the graph. For example, with point discontinuity, there could be a hole and a point on the same x-value, although the intended height is where the hole is, the real value is where the defined point is. Lets look at a picture:

Looking at the point discontinuity at x=-2, we see that the LIMIT as x approaches -2 is 1, however the VALUE as x approaches -2 is actually 3.

How do we evaluate limits numerically, graphically, and algebraically?

To evaluate limits numerically, we must create a table that has rows for the x-value and the f(x) value (on the left side). The number we are trying to reach (which is the limit) will be in the middle. On the left side of the approximated limit, we will have numbers that are slightly less than the x-value and on the right side we will have numbers slightly larger than the x-value. Trace these x-values in your graphing calculator and record the f(x) values. Once you have found the limit, write out the limit statement based on what you've found. lim x-># f(x) = # and verbally this would read as "the limit as x approaches # of f(x) is equal to #" Be sure to include a short explanation as to why the limit can or cannot be reached.
 To evaluate limits graphically look at the picture of the graph. Now that you see the graph, place two fingers--one on the left side of the limit and one on the right side of the limit--then trace your fingers (follow the line) to have them meet where the x-value is. If your fingers meet, then that is where the limit is! If the fingers do not meet, the limit does not exist. And this can be due to unbounded behavior, jump discontinuity, or oscillating behavior.
To evaluate limits algebraically use direct substitution, dividing out/factoring method, or the rationalizing/conjugate method. For direct substitution, all you do is plug in the number that "x" approaches into the equation. NOTE: When using direct substitution, there are 4 types of answers we can get:
1. a numerical answer
2. 0/# -this is just zero
3. #/0- this is undefined which means the limit does not exist
4. 0/0 -indeterminate form (this one is special...meaning we must use another method!)
For options 1-3, this means we are DONE with the problem, but if we get indeterminate form (option 4), then we must use the dividing out/factoring method or the rationalizing/conjugate method.
Dividing out/Factoring-This process is done by factoring the top and bottom and canceling out any items that we can. By doing this, we cancel out the zero in the denominator. Only then, can we use direct substitution to find the limit.
Rationalizing/Conjugate- This process is where we change the sign--positive or negative--in the middle of the 2 terms. We use the conjugate wherever there is a radical in the expression, so that could be either the denominator OR numerator.
1. Multiply the conjugate to both the top and bottom of fraction
2. Foil
3. Simplify by canceling out common factors (be careful!)
4. Use direct substitution to find the limit

Works Cited: Bridge-Wikimedia Commons
Continuous function-http://www.conservapedia.com/Continuous_function
Discontinuity pictures-http://faculty.wlc.edu/buelow/calc/nt2-4.html
Oscillating graph picture- http://webpages.charter.net/mwhitneyshhs/calculus/limits/limits.html
Youtube Video- http://youtu.be/392XQbIl8TQ
Point discontinuity-http://www.zweigmedia.com/RealWorld/calctopic1/contanddiff.html