Continuity means that the function will be predictable, meaning it has no undefined points, no breaks, no "holes", and no jumps. To have a continuous function you must make sure that you can draw this function without needing to lift up your pencil from the paper. You also have a clue that you have a continuous function when the limit (INTENDED height) of a function is the same as its value (ACTUAL height).
You know that a function is at discontinuity when the limit is not the same as the value because the function has an undefined point, a break, a "hole", or a jump. At discontinuity, it is impossible to draw your function without needing to lift up your pencil from the paper at some point.
You must keep in that there are two families of discontinuity: Removable and Non-Removable.
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. It means that you are getting VERY CLOSE to a certain height as you are getting VERY CLOSE to a certain x-value. A limit differs from a value because as it is only the INTENDED height of a function, the value is the ACTUAL height of a function. For a value you do not just get VERY CLOSE to a certain height, you ACTUALLY GET there. If the function has the same limit and value it is a continuity, there is no ambiguity to it, but if the function has different limit and value then you know it must be a discontinuity.
Consider that there are two families for discontinuity. The limit exists only in the removable discontinuity, also known as the point discontinuity(it is the only one in that family), because the same height is intended to be reached from both the left and the right sides of a function. You put your fingers on both ends of the function,left and right, and you bring them together to meet at the same place, even if the place isn't there (point/"hole"). If they do not meet at he same place, then there is a non-removable discontinuity and the limit DOES NOT EXIST. The limit does not exist in 3 types of the non-removable discontinuity family. One is called jump discontinuity (the limit DNE because of different left and right); another is called infinite discontinuity (the limit DNE because of unbounded behavior); and finally there is oscillating behavior (the limit DNE because there is not definite point that is reached at a certain x value).
3. How do we evaluate limits numerically, graphically, and algebraically?
a) Numerically- For instance, the picture below shows the table needed to find limits numerically. If finding the limit as x approaches 3 (#9), we would plug the function of #9 into our calculator at "y=", we would hit "graph" and "trace" for the numbers getting VERY CLOSE to the x-value 3. Our table like shown below will help us organize our information. In the middle box for the x value (on top if the ?) we would put the 3. The 3 boxes on the left should be the x values getting VERY CLOSE to 3 from the left side--2.9, 2.99, 2.999--and the 3 boxes on the right should be the x values getting VERY CLOSE to 3 from the right side--3.1, 3.01, 3.001.
https://drive.google.com/file/d/0B4NSkh2FgPbXR2RSOUU0cFVvWWs/image?pagenumber=6&w=800
b) Graphically- You just take a look at a graph like the one shown below and write down what you see happening in terms of the limit at each x value you need to take a look at.
https://drive.google.com/file/d/0B4NSkh2FgPbXR2RSOUU0cFVvWWs/image?pagenumber=5&w=800
c) Algebraically- There is direct substitution, in which you plug in the x value directly to the function to get the limit. You can get 4 different types of answers. There is numerical answer (ex. 3)...YAY we are done! There is a answer with a 0 divided by some number, which equals 0...YAY we are done! There is an answer when a number is divide by 0, which we know is undefined, and therefore the limit DNE...YAY we are done! Finally there is a 0 divided by a 0, which means that it is "not yet determined" and you must keep working on another method to figure it out. Note: remember 0/0="hole"
Another method is dividing out/factoring. IF you try direct substitution and you get a 0/0, then you may move on to factoring out a polynomial, from whichever numerator or denominator as needed, and get rid of the zeroes. Then you use direct substitution on whatever is left after getting rid of the zeroes.
The last method is called rationalizing/conjugate because if you have a square root in either the numerator or denominator you must multiply by its conjugate to cross out the zero. Once again you must consider this only if you have definitely tried the direct substitution method first and you got 0/0.
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