BQ# 6: Unit U Concept 1-8: Defining Limits

1. Continuity is when a function can be drawn without lifting your pencil from the paper. A continuous function also has no breaks, jumps, or holes.

 

 Although there is a hole in the function, the limit is there. A limit is an intended height, and this function is continuing on going through that intended height. The function is continuing on with it's pattern. The function is continuous. 

A discontinuity is a point in which the function stops and has a change in behavior. A jump discontinuity is when the function from the left and the right meet but have different y values, it changes and so you see two different segments. An infinite discontinuity is when there is a vertical asmyptote which leads to unbounded behavior and this causes the function to be discontinuous. An oscillating discontinuity with when the function has a very wiggly behavior and is very inconsistent, this means there is no definite point so the limit does no exist which means it is a discontinuity. 

2. A limit is the intended height in a function. A limit will always exist unless there is a jump discontinuity or infinite discontinuity or an oscillating graph. Besides these few points, their should be a limit at every single point you are on. If you happen to land on one of these points, the limit will not exist. The differences between a limit and a value is that a limit is the INTENDED height and the value is the ACTUAL height of the function.

3. When we evaluate a function numerically, we take a look at the table. We use tables to identify discontinuities. Evaluating a function graphically is the easier way you can look for limits/discontinuities. You can just take you fingers and bring them together to see what's going on on the graph. You will be able to see right away whether it is continuous or if there is a discontinuity. There are several steps in which you would have to do in order to evaluate a limit algebraically. You must first do substitution. You will substitute in the limit it is asking for and if you have a regular fraction or 0/#  or #/0 than that will be your limit. If it is 0/0 than it is indeterminate and you have to continue on by either factoring the equation or rationalizing/conjugate. From there you have to cancel out to remove the hole in the equation. Once things have canceled then you are able to use substitution again to find the limit, and that's how you evaluate a limit algebraically.