BQ#6: Unit U

1. What is continuity? What is discontinuity?

Continuity means that the graph is predictable. Also it has no breaks, holes, or jumps, and it can be drawn without lifting the pencil off the paper.

Point Discontinuity

Oscillating Behavior

Jump Discontinuity

Infinite Discontinuity

 

 

            Discontinuities can be seen in graphs when it has point, oscillating, jump, and infinite discontinuities. Within these discontinuities there are two groups, removable and non-removable discontinuities.  A type of removable discontinuity is a point discontinuity. This means that there's a hole in the graph(as seen in the first graph). Non-Removable discontinuities consists of jump discontinuity, oscillating behavior, and infinite discontinuity. Jump discontinuity means that if we trace the graph with our left and right finger on the graph and get to different points(third picture). It is important to know that the jump can have two open circles, one close/open, but they must never be two closed circles. Oscillating behavior is also part of non-removable discontinuities. In oscillating behavior the graph is very wiggly, or unpredictable. In this case the limit dne because you can't really cant tell where the graph's next destination will be. Infinite discontinuity is the last part of non-removable. This is when there's a vertical asymptote leading to unbounded behavior. Unbounded behavior means the graph is either going up infinitely or down infinitely (last picture)

 2. What's a limit? When does a limit exist? When does a limit DNE? What's the difference between limit and value?

                A limit is the intended height of a function. A value is the actual height of a function. A limit exists when the intended and actual height are the same (value) . A limit is the value that a function or sequence "approaches" as the input or index approaches some value. A limit does not exist when the left and right are different(jump), unbounded behavior(vertical asymptote), and oscillating behavior. 

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

To begin numerically means on a table. So with this in mind we evaluate a limit numerically by making an x and y table with seven spaces. Then we write the limit in the middle box. Afterwards we come up with the x values by adding and subtracting 1/0. Once we have that we plug in the function and trace the numbers that we have on the table (x-values). Once we plug in the limit if the calculator does not give us an answer then it is undefined. For algebraically there are three methods that can be used. The first method is substitution. In this method we just plug in the limit and solve. There are three answer that we can get which are a number, a zero divided by a number which is zero, a number divided by zero which is undefined, and 0/0 which is indeterminate form. If we get indeterminate form we have to use the dividing/factoring method or rationalizing/conjugate method. In the factoring method we have to factor the numerator and denominator and cancel like terms to remove the zero. Then with the simplified expression, use substitution to find the answer. The rationalizing/conjugate method is when you multiply the radical with its conjugate. Remember to not multiply the non-conjugate because something would hopefully cancel. Once something does cancel, you use substitution to get your answer. 

 

 Factoring Method

 

substitution method 

 

 Rationalizing Method