Unit U Big Questions

• What is continuity? What is discontinuity?

A continuity is something that goes on and on without getting interrupted by other things. But for discontinuity, it means the opposite. In calculus, a continuous function has no break, no holes, and no jumps. However, the discontinuous function contains some interruptions due to the jump, break, and/or hole.
Example of Discontinuity:

From www.sagemath.org

Example of continuity:

From tutorial.math.lamar.edu 

• 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. It exists when both the right hand limit and the left hand limit must be the same. A limit does not exist if the graph has a break, including jump, oscallating, and infinite discontinuities. A value is the number that always belong and appear in the graph (it can be defined). But a limit can sometimes exist and sometimes cannot exist. A limit can still exist if your ultimate destination is a hole in the graph.

• How do we evaluate limits numerically, graphically, and algebraically?

We evaluate limits numerically by creating a table that has x-value and f(x) value on the left hand. The first row will show the limit from the left and the limit from the right. Using the number that the limit as x approaches, it will be the value that we need to look for. Next, we need to write the limit statement mathematically (lim x-># f(x) = ) and the limit statement that contains the limit as x approaches _______ of _______ is _______ and a short sentence explain whether or not the limit can be reached (include a reason).
We evaluate limits graphically by looking at the given graphs and answer the questions. An efficient way to do it is using two finger, one goes from the left and one goes from the right side. If the fingers do not meet, the limit does not exist. If the graph has a point discontinuity, the limit still exist but if it has jump, oscallating, and infinite discontinuities, the limit does not exist at all.
We evaluate limits algebraically by using direct substitution, dividing out/factoring method, or rationalizing/conjugate methods. For direct substitution, it is very easy because just use the number that the limit as x approaches to plug into the equation. There are 4 results that we can get from that method: 1. a numerical answer, 2. 0/# -this is 0, 3. #/0- undefined which means limit does not exist, and 4. 0/0 -indeterminate form ("not yet determined"). If one of the first 3 answer appears, it means that we are DONE. However, if the answer is indeterminate form,  we have to use dividing out/ factoring method. It is similar to the problem that ask to "Simplify" a fraction. By factoring out both the numerator and denominator and cancel common terms to remove the zero in the denominator, we can use the direct substitution after to find the limit. Rationalizing/Conjugate method is what we already learned throughout the year. The conjugate is where we change the sign in the middle of 2 terms. Using conjugate of the denominator or numerator depends on where the radical is. Step by step: 1. Multiply the conjugate on top and bottom, 2. simplify by foiling (does not have to foil the one that is non-conjugate), 3. elminate from the top and bottom by canceling out common factors, 4. base on the simpliest form of the equation, use direct substitution to find the limits.