BQ #6: Unit U: Continuities, Discontinuities, Limits, & Values

1. A continuity goes on forever, is predictable, can be drawn with a single long pencil stroke (without having to lift the pencil) and has no breaks, jumps, or holes. Because it is called continuous, that means that the graph is one straight line, like in slope-intercept form, which continues on in the same pattern.
A discontinuity is just the opposite of continuous in that it can have holes, jumps, and breaks and can't be drawn with one long pencil stroke. They're unpredictable and all over the place. There are two families of discontinuities: removable and non-removable.

In Removable, there is only one type: a point discontinuity which has a hole. Under Non-Removable, there are three types: jump, oscillating behavior, and infinite. A jump discontinuity is different from both the left and right side and can sometimes be broken. Oscillating behavior is really wiggly and resembles a seismograph. Because it gets so crazy in the middle, it's difficult to determine. Infinite occurs when there is a vertical asymptote which is caused by unbounded behavior, which means that it increases or decreases without bound towards (negative/positive) infinity.

2. A limit is the intended height of a function whereas value is the actual height reached. A graph can have infinite limits. However, a limit is reached when both the right side and left side of the graph reach the same destination, which, in that case, is also the value. To determine whether a limit is reached, we trace the graph until we know that the limit is or isn't reached. This is also how we graphically evaluate a function's limit. A limit is also different from a value in that it can't exist is when the left and right are different (jump discontinuity), when there's unbounded behavior (infinite discontinuity) and during oscillating behavior.

3. Limits can be evaluated numerically, graphically, and algebraically. When evaluating limits numerically, we use a table to help us see how close we get to the asymptote without ever actually touching it. As already mentioned, when evaluating graphically, we trace the left and right sides of the graph to see where we end, whether the left and right sides of the graph actually meet their destinations. There are several ways to evaluate a limit algebraically. It's important that the first thing we should do is apply the direct substitution method, which is basically plugging in the number to see if its works. If we get a numerical answer, or 0, or DNE, then we know that it works. If we happen to get 0/0 (indeterminate), then we have to try something else. Other methods to try if we get indeterminate are the dividing out/factoring method, rationalizing/conjugate method, and limits at infinity (long method).