What is continuity? What is discontinuity?
A continuity is something that goes on and on without getting interrupted by other things. Discontinuity means that it goes on but it gets interrupted by jumps, holes, breaks.
Example of Discontinuity: Example of continuity:
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 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. The difference between a limit and a value are that a value is the number that always belong and appear in the graph. A limit can sometimes exist and sometimes cannot exist. A limit can still exist if there 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 side. We use the number that the limit as x approaches, which will be the value that we need to look for. 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.
We evaluate limits graphically by looking at the given graphs and answer the questions. If the sides do not meet, the limit does not exist. If the graph has a point discontinuity, the limit still exist. If it has jump, oscillating, 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, use the number, that the limit as x approaches to, and plug it into equation. There are 4 results 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. If one of the first 3 answer appears, it means that we are done. If the answer is indeterminate form, we have to use dividing out/ factoring method. 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. 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. 1. Multiply the conjugate on top and bottom, 2. simplify by foiling 3. eliminate from the top and bottom by canceling out common factors, 4. base on the simplest form of the equation, use direct substitution to find the limits.
