Photocredits: http://www.mathsisfun.com/calculus/continuity.html
In order for a graph to be continuous, first of all, must be predictable. Secondly must not have any jumps, holes or vertical asymptotes. Lastly, must be drawn without lifting the pencil off the paper. All of these elements can be seen in the pictures above. Now discontinuities are the exact opposite of this. But do note that there are two different types of discontinuities: removable and non-removable. A question frequently asked is why are there two groups and that my friends is because removable are when limits do exist and non-removable discontinuities do not exist. One removable discontinuity is point and the limit here does exist because although it does not reach the intended height it does reach a certain height the actual height and that is seen through the value. The three non-removable discontinuities are jump, oscillating and infinite. A limit cannot exist in jump because there are differences in left and right graphs and they do not meet. A limit does not exist at oscillating because, well, it cannot be found. The graph is too out of control to actually find where the limit exists. In an infinite discontinuity a limit does not exist due to unbounded behavior.
Photocredits: Unit U SSS Packet by Mrs. Kirch
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 a function. A limit exists when a graph is continuous or as previously stated when, no jumps, holes or vertical asymptotes are present. A limit does not exist when there are left and right differences (jump), when the graph is wiggly (oscillating) and when there is unbounded behavior (infinite). All of these stated are elements of non-removable discontinuities. Lastly, there is a BIG difference between a limit and value. A limit is the intended height of a function, BUT the value is the actual height of the function. When solving equations it is extremely important to know the difference between the two knowing it will affect your answer (ya know, whether you get it right or wrong. OBVI you wanna get it right soo...know the difference!).
3. How do we evaluate limits numerically, graphically, and algebraically?
Numerically- To solve limits numerically a table would be necessary and key to help you solve. The limit statement that is said verbally. Is "the limit as x approaches # of (blank) is #. Or as seen in the image below
Photocredits: Unit U SSS Packet by Mrs. Kirch
Graphically-
Photocredits: Unit U SSS Packet by Mrs. Kirch
As seen above a limit was solved graphically. It is vital to know the difference of the value and limit here because solving it graphically makes it more difficult. I through M are all limits for I and M since they do not exist a b/c is necessary in order to fully understand why the limit DNE. Sometimes, because a limit does not exist, it does not mean that a value is present. So when solving a limit graphically it is HIGHLY important to pay attention to all the little details, they are very important.
Algebraically- There are three ways to solve a limit algebraically those being: direct substitution method, dividing out/factoring method, and rationalizing/conjugate method.
FIRST method to try out would be the substitution method in which you substitute all the variables with the give number that x approaches in the limit statement. If it turns out to be undefined the next step to trying to solve it out is using the dividing out/factoring method. In this method we factor out any long equations and then find common factors and divide them out. With what is left from there we substitute the given number x approaches in the limit statement given and if we get an answer AWESOME-SAUCE and if we don't we go for our last and final method that being: the rationalizing/conjugate method! In this method you multiply by the conjugate in the denominator to eliminate some things and make life much easier, this method is the ONE. And most of the time works out fine so whenever the other two methods do not make your life easier, come to this one! If more help is needed in understanding the last method, look at the image below:
Photocredits: Unit U SSS Packet by Mrs. Kirch
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