Big Question #1,2,3

1. What is Continuity? What is discontinuity?

Continuity: Continuity means that the graph may go on and will have no interruptions. Continuity graphs will have no holes, jumps or any type of breaks. 

Example: As shown the picture is continuous and has absolutely no interruptions.

Discontinuity: A discontinuity graph means that there has to be interruptions in the graph such as a jump, hole or breaks.

Example: This image shows that there is a jump discontinuities and holes present in the graph.

Pictures from : http://www.zweigmedia.com

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

• What is a limit? 

       - The limit is the intended height of the function. It is the y-value where the x axis lies. 

• When does a limit exist? 

      - The limit will exist when the right and the left reach the intended height. It has to be the same when reaching the left or right. The limit may also exist at a hole (not actual value).

• When does the limit not exist? 

     - The limit will not exsist if there is an interruption such as jump, oscillating, or infinite discontinuity. 

• What is the difference between a limit and value?

     - The limit is the intended height, while the value is the real height of the given function. 

Example: Image shows the jump discontinuity, 

example: This picture shows the jump continuity and the left or right approach for the limits (which DNE because of Jump) 

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

• Numerically: To solve numerically, we first have to set a table, which in hte first row it will show the limit from both the right and the left. On the second row it should contain f(x) and the value of the limit is being approached. Then state the limit statemnt, mathematically and verbally. 

example: Shows a table in which how the limit approaches its value.

• Graphically: To solve graphically we use the method of two fingers(left and right side of graph) in which we place on the graph. If the fingers meet at each other then the limit exists  If the fingers are do not meet each other then limit does not exist. If the limit does not exist then it must be because of jump, oscillating, and infinite.
• Algebraically: To solve it algebraically we use the substitution method. To use the substitution method we simply plug in the numbers into the equation, while using this method we might get a numerical answer, 0/# which will be zero, a #/0 means the limit is undefined. And if the answer is 0/0 then it is indeterminate. If it is indeterminate then we use the dividing and factoring out method, which we factor both the numerator and denominator and cancel out similar terms, then use direct substitution. The other method is the rationalizing and conjugate method by using this method we multiply the conjugate change the sign, foil, and simplify.