BQ 6: Unit U - Concepts: 1-6

Continuity & Discontinuity ❤︎

Continuity happens in a continuous function. a continuous function is a function that is predictable, which means it will go wherever we assume it will go, it has no breaks no jumps no holes and you may draw the graph without lifting your pencil from the paper. Graphs are continuous on intervals. Continuous functions make good bridges. A function is also continuous if the value and the limit are at the same point. 

Discontinuity happens when the graphs do contain breaks and or holes. Discontinuities are broken into removable and non-removable families. in removable continuities there are point discontinuities these graphs are also known as holes. In point discontinuity the value and the limit can sometimes be the same. In non-removable discontinuities there are jump discontinuities, oscillating behavior and infinite discontinuities. There is a break in Jump Discontinuities, In oscillating behavior the graph is wiggly so we cannot tell where the point are and in infinite discontinuities there is a vertical asymptote which leads to unbounded behavior.

Limits∞ 

A Limit is the intended height of a function and the limit exists when the value and the limit are the same, it also exists in continuous functions as well. A limit does not exist when the value and the limit are different. Limits do not exist in a jump discontinuity due to the fact that the points arrive at different locations from both the left and the right of the graph, it doesn't exist in oscillating behavior due to the fact that the graph is very wiggly and we cannot tell where the points lye on the graph & it doesn't exist at infinite discontinuity because of unbounded behavior due to the vertical asymptotes. The difference between a limit and a value is that the limit is the intended height and the value is the actual height.   

Evaluating Limits ☁︎

We evaluate graphs numerically by making a table and start off with the number we are heading toward and as we go to the left we subtract 1/10th and as we go to the right we add 1/10 and as we do this we can see that we are getting closer to our value as we move to the middle. We evaluate graphs algebraically by using either the dividing out/factoring method & the rationalizing conjugate method. Finally we evaluate a graph graphically by putting one finger on the right side of the graph and one finger on the left side and we slide our fingers along until they meet and we see where they meet and if they do in fact meet at a certain point on the graph to determine whether they are removable or non-removable.