BQ #6 - Unit U

1. What is a continuity? What is a discontinuity?

A continuity--or better yet a continuous function-- is predictable and does NOT have breaks, holes, nor jumps. Continuous functions can be drawn without lifting the pencil/pen off the paper. On the other hand, a discontinuity is the opposite and are in two families: Removable and Non-Removable Discontinuities. Under Removable discontinuities exists point discontinuity (known as a hole), like in the images below. [Notice, point discontinuities can exist with or without a filled circle]

(http://www.mathwords.com/r/r_assets/r88.gif)

http://images.tutorvista.com/cms/images/113/removable-discontinuity.png

Under Non-Removable Discontinuities exists three more. Jump behavior one of them and is when there is a break or a jump in the function. Here, the left side or the function is different from the right and both sides won't meet at the same f(x) value. See the image first below. Jump discontinuities can exist with either 1. two open holes or 2. one opened and the other closed, but never both closed circles. Another disconinuity under this category is oscillating behavior, which is described as "wiggly" like in the second image. The third and final is infinite discontinuity. It contains--or is composed--of a vertical asymptote that causes unbounded behavior as it will never reach an actual value. See the third image below.

1.

(www.wikimedia.org)

2.

(www.cwladis.com)

3.

(www.milefoot.com)
  

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. The limit will exist at any continuous function as well at Removable Discontinuities, or point discontinuities. The limit DOES NOT EXIST at any of the Non-Removable discontinuities. With jump discontinuities the left and right side of the function, approaching at a specific x value, will not have the same height; therefore there is no limit. With oscillating behavior, the limit won't exists as it doesn't approach any single value. In infinity discontinuity, the vertical asymptote causes the unbounded behavior to occur. Both sides will ATTEMPT to reach a value, going the same direction or opposite, but never will. As f(x) will approach infinity, the limit is unbounded since infinity is not a number.
     As the limit is defined as the intended height of a function, the value is defined as the actual value of a function. "Intended height", in my perspective, can be interpreted as "the calculated/predicted position or point." The value is what determines where the position/point is actually at. The limit and value can 1. both exist and be the same or 2. can both exist and be different (in some point discontinuities). There are functions in which 3. the limit does exist but there is a value of undefined (in some point discontinuities) or 4. where the value exists but there is no limit (in some jump discontinuities)
1.

(www.math-spot.com)

2.

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

3.

(https://images-blogger-opensocial.googleusercontent.com)

4.

(www.mathwords.com)

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

     To evaluate limits numerically, a number table is used. The value of x is placed in the middle with three other values increasing to the right and decreasing to the left. The x values would increase or decrease by decimal values to the tenths, hundredths, and thousandths places, like in the image below. As f(x) can be predicted, the limit is simply approached, however it can SOMETIMES be reached (in continuous functions only).

(from Mrs. Kirch's Unit U SSS Packet)

     To evaluate limits graphically, view the graph and follow the left and right side of the function--use your fingers if you have to--to a specific x-value. If both sides reach the same height/location, then the limit exists. If both sides do not reach the same height/location, then the limit does not exist. The image below is an example of the left and right side approaching the same x-value.

(www.wyzant.com)

     To evaluate limits algebraically, use the direct substitution method and plug in the x-value into the equation. If a numerical answer, 0 over a number (results as 0), or a number over 0 (results as undefined), then the evaluation is complete. However, if a fraction occurs as 0/0, then it is indeterminable and the evaluation continues. Only then one must use the dividing/factoring method or the rationalizing/conjugate method. In the dividing/factoring method, the numerator and denominator are factored in order to have something cancel out from the bottom and to avoid the denominator equaling 0. In the rationalizing/conjugate method, the equation is multiplied by the conjugate of either the numerator or denominator with the radical. 

     If x is ever approaching +/- infinity, then it is also an indeterminable form; however, there is a different way to solve this form. One must divide every term (in the numerator and the denominator) by the highest power of x found in the denominator. Infinity is then substituted in but REMEMBER: 1) a number value divided by infinity is 0; 2) equations without an x in the denominator do not have a limit due to unbounded behaviors; and 3) if the numerator has the same highest degree as the denominator, the limit is based on the ratio of the coefficients.