A continuity is a continuous function that is predictable has no holes, no breaks, and no jumps. You can also draw it without lifting your pencil from the paper. Another key characteristic is that the value equals the limit. An example of a continuous function would be this below:
A discontinuity is the complete opposite to that of a continuous function. There are two groups of discontinuities: the non-removable and removable discontinuities. A point discontinuity involves the graph having a hole, meaning that limit does exist but because of the hole the value is undefined. But if there were to be a black dot below it or underneath it that would give us the value. A point discontinuity is within the removable discontinuity. Moving onto the non-removal discontinuities the first one is called a jump discontinuity in which involves different left right endings, end in different locations.It's important to keep in mind that both circles cannot be closed or it is not a function, also if both circles are open the value of the function is undefined. The next type is an oscillating discontinuity which is basically a wiggly graph, and is rarely seen. The last type would be infinite discontinuity which because of the vertical asymptote the graph exhibits unbounded behavior. Unbounded behavior simply meaning that increases or decreases without bound. Each one clearly portrayed in the image below:
2. What is a limit? When does a limit exist/not exist? What is the difference between a limit and a value?
A limit is the intended height of a function. A limit does not exist when dealing with the three non-removable discontinuities: jump, oscillating, and infinite. In a jump discontinuity the limit would not exist because both lines do not meet at same place (an intended limit). This is seen when we evaluate the limit by placing our fingers on a spot to the left and to the right of where we want to evaluate the limit, if our fingers do not meet our limit does not exist. The next type would be infinite discontinuity in which the limit does not exist because of unbounded behavior, but value does exist at one value end of the one sided limits. The last type being an oscillating discontinuity in which limit does not exist because of oscillating behavior. A limit does exist when dealing with a continuous funvtion because the limit and the value equals one another. A limit is the intended height of the function and value is the actual height of a function (where black dot lies).
The image below shows limits in which the limit/values that do or do not exist:
(Mrs. Kirch)
3. How do we evaluate limits numerically, graphically, and algebraically?
We evaluate limits numeracally on a table shows how we gradually get closer and closer to a limit and value. This is done by making a table and at center placing the limit to get the limit from left we subtract 1/10 and to get limit from right you add 1/10. And to find the values we plug the function into our calculators and trace for value we are looking for. Next graphically is finding the limit bases on use of graphing calculator by typing in function, going to graph, hitting trace, and just tracing for the value we want. Or by being given a graph in which we simply evaluate the limit by placing our finger on a spot left and to the right of where you want to evaluate the limit. When it comes to algebraically we use three different forms: direct substitution, dividing out/factoring method, and rationalizing/conjugate method. Direct substitution involves plugging in the number x is approaching, and if it results in a numerical answer you are done, if it results in 0/# answer is zero and you are done, if we get #/0 then it's undefined meaning limit does not exist (here we must state a reason as to why doesn't exist), if we get 0/0 it's un determinate form meaning we have to use another method. Another method like the dividing out/factoring in which we factor both the numerator and denominator and cancel common terms to remove the zero in the denominator, then proceed by using direct substitution with the simplified expression. The other method is known as rationalizing/conjugate in which: if it's a fraction we multiply by the conjugate ( only the sign in the middle of both terms changes), depending where the radical is you take the conjugate of where it is, that either being in the numerator or denominator. Then we simplify by foiling, leaving the non-conjugated denominator or numerator factored. Next proceeding by canceling out like terms, leaving us with a simplified expression. Finally using direct substitution with the simplified expression. One huge thing to always do is to try direct substitution first, and if doesn't work out proceed with the other two methods.
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