Unit U Big Questions

1. What is continuity? What is discontinuity?
Continuity demonstrates that a function is predictable and has no breaks, holes, or jumps. A continuous function can be drawn with a single, unbroken stroke of a pencil. You can tell if a function is continuous by imagining that you are driving a car on the line. If the imaginary car meets anything that makes the car fall off the line such as a hole or jump, the graph is discontinuous. Discontinuity or a discontinuous function has either a jump discontinuity, oscillating behavior, or a vertical asymptote. In reference to the imaginary car, a discontinuity would make the car fall off the line. 

2. What is a limit? When does a limit exist? When does a limit 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 graph reaches the same height from the left and right side. This can be tested by placing two fingers on the graph and tracing the line towards a certain point to see if your fingers meet. If they meet, the limit exists at that point. A limit doesn't exist when your two fingers do not meet on the line or in other words when the left and right behavior of a graph do not match and when their is unbounded or oscillating behavior. The value is the actual height of a graph, while a limit is the intended height. 

3. How do we evaluate limits graphically, algebraically, and numerically? 
We evaluate a limit graphically by determining left and right behavior. We do this by using two fingers to trace to a certain point and of our fingers meet at the same point, the limit exists. If they do not, then that is most likely due to a hole or jump and the limit doesn't exist. We evaluate limits algebraically by using either direct substitution, dividing out/factoring method, or rationalizing/conjugate method. These methods can all result in indeterminate form which is 0/0 and so we have to figure out which  method works best. Direct substitution is substituting the limit into the equation.  Dividing/factoring out method is factoring the numerator and denominator and canceling terms then substituting in the limit. Rationalizing/conjugate method is multiplying by the conjugate of the term with the radical in the function. We evaluate limits numerically by creating a table with values that get closer and closer to a certain value on the x axis and notice that the graph of f(x) get closer to a value on the y axis. Our table consists of x values that approach the limit from the right and left and after plugging the function into our graphing calculator we can trace to the x values and record the y values. With our table we can determine if the limit can be reached or not. If the f(x )value doesn't match the limit on the table, then the limit cannot be reached.