Unit U Big Questions

Unit U Big Questions

1.) What is continuity? What is discontinuity? 

Continuity means continuous. That means that when we draw a function we are able to draw it without lifting our pencil from the paper. In other words a a function does not have any breaks, no jumps. and no holes. This is how it is defined in calculus. The following functions are continuous:

Discontinuity refers to when a function is not continuous. This means there are jumps, breaks, or holes in the function. In calculus there are 3 types of discontinuities broken down into families. The two families are:

1. Removable discontinuities :     - Point discontinuity (aka a Hole) ; if f(x) has a hole this would be known as a point discontinuity  
2. Non-Removable Discontinuities:      - Jump discontinuity; if a function has a break or snap it is called jump discontinuity                                                         

                                                                     - Oscillating Behavior; a very wiggly function
                                                                     - Infinite Discontinuity ( aka unbounded behavior and occurs                          where there is a vertical asymptote)
 
               

2.) Limits

    What is a limit?

                        - A limit is the intended height of a function. As we go along the X-axis closer and closer to a a certain value, the graph f(x) gets closer to a value on the Y-axis. This y-value is referred to as the limit which is the intended height. 

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

                 * watch the video and find out :)

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

• Numerically means on a table. So we evaluate limits numerically using an x and y table. We know that a limit is when x approaches a certain value from the left and from the right. So to do the table we put the number that it is approaching in the center and values from the left and from the right the the table below: 
  
• Graphically means looking at a graph. For this we plug in the f(x) equation into the graphing calculator and we use the given picture of the graph. We can identify wherever there are any discontinuities and in doing so we can evaluate the limits just as the picture below:
  
• Algebraically means solving using algebra skills. To do this there are three methods as our options. We have direct substitution, dividing out, and rationalized. The first method we use for all functions is direct substitution. This means we simply plug in the value x is approaching into the function equation and the result is our limit. If we ever get a numerical answer then we are done. If we get a 0 as our answer then we are also done. However if we get  and undefined value then the limit does not exist and if we get 0/0 then this is indeterminate form which means we have to use another method. 

Citations:
http://www.analyzemath.com/calculus/continuity/continuous_functions.html

http://thetwomeatmeal.wordpress.com/2010/11/04/continuous-maps-and-homeomorphisms/