A continuous function is at most predictable. It consists of no jumps, no breaks, and no holes. You can draw a continuous function without lifting up your pencil. A discontinuity has two major families: removable discontinuities and non-removable discontinuities. The removable discontinuities only consists of a point discontinuity where a hole exists. The non-removable discontinuities are jump discontinuity (different left/right), oscillating behavior (wiggly), and infinite discontinuity (unbounded behavior and vertical asymptotes).
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 the function. A limit exists at removable discontinuities, which also refers to point discontinuity or hole. Because from left and right, it is the intended height for both sides and therefore, the limit does exist. A limit does NOT exist at non-removable discontinuities, which includes jump discontinuity, oscillating behavior, and infinite discontinuity. A limit does not exist at jump discontinuity because of different left and right. The limit does not exist at oscillating behavior because it is wiggly and has no single value since it does not approach it. The limit does not exist at infinite discontinuity because of unbounded behavior due to the vertical asymptote.
While the limit is the intended height of the function, the value is the actual height of the function. In the picture below in the first graph, the limit exists but the value does not. It is undefined since f(c) is a hole. In most cases, the value is undefined when there is a hole. In the second graph, the limit exists but the value exists somewhere else. Although the intended height is known, the actual height has already been defined. In the third graph, the limit does not exist because of jump discontinuity. The value, though, still exists. It exists at only one of one side limits or at the closed circle.
3. How do we evaluate limits numerically, graphically, and algebraically (VANG)?
Algebraically
To solve limits algebraically, there are three different methods: direct substitution method, factoring method, and conjugate method. In direct substitution method, you simply plug in the number given and see what you get. In the factoring method, you have to factor out both the numerator and denominator. Then, cancel common terms in order to remove the zero in the denominator. In the conjugate method, you simply rationalize the either the numerator or denominator depending where the radical is. When using these methods, always try direct substitution method first! If you plug in x and get 0/0 (indeterminate form), then you would either use the factoring or conjugate method.
Numerically
To find limits numerically, you will need to make a table. Before finding the limit, start from left and right on both ends of the table. Add/subtract a tenth of what is given. Then, write down the numbers that will eventually get smaller and smaller. For example, 2.9, 2.99, and 2.999.
Graphically
To determine the limit graphically, place your fingers on the graph to the LEFT and RIGHT of where you want your limit to be evaluated. If your fingers do not meet, then the limit does not exist due to different left/right, wiggly, or unbounded behavior because of vertical asymptotes.
REFERENCE:
Mrs. Kirch's SSS packet
