A continuous function means you can draw it without your pencil lifting from the paper. It is predicatble meaning you know whats going to happen. There are no breaks, jumps, or holes. It can be drawn with a single unbroken pencil stroke. ex: lim as x approaches c of f(x)=f(c)- the limit and value are the same. On the other hand a discontinuous function is UNpredicatble. There are 4 types of discontinuities. A point discontinuity, which is also known as a hole, a jump discontinuity(different left and right ), oscillating behavior (wiggly), and an infinite discontinuity which results in unbounded behavior caused by a vertical asymptote. These continuities from two families, non removable, and removable discontinuities.
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 the function. A limit exists as long as you reach the same height from the left and the right. A limit does NOT exist if the right and left do not meet.
In the above picture the limit DOES exist because from both the left and right it approaches the same intended height.
In the above picture the limit DOES not exist because of a jump discontinuity and also because the left and right do NOT reach the same intended height.
A limit is the intended height, which means it can or cannot reach it, and the value is the ACTUAL height, or what it reaches.
3.How do we evaluate limits numerically, graphically, and algebraically?
To evaluate limits algebraically you can use three " shortcut" ways. There is the direct substitution way which is basically plugging in the x into the function. When doing this method you can produce 1 of 4 answers. You can get a numerical answer which means you are done. You can get a 0/# answer which means you are done. You can get a #/0(undefined) and you are done. Finally you can get 0/0, or indterminate form which requires more work. If the first method does not work you may use the dividing out/ factoring method. In this case you factor out the numerator and denominator and cancel out common terms to remove zero in the denominator. Then , use direct substitution with the simplified expression. Lastly, there is the rationalizing/conjugate method. If it is a fraction , then multiplying the top and bottom by a conjugate would have helped. The conjugate is where you change the sign in the middle of the term. You use the conjugate of wherever the radical is.
To evaluate a limit graphically we look at the graph and from the left and right with our fingers you trace the limit. This means it has to be the same from the left and the right. Make sure to identify the discontinuities.
To evaluate the limit numerically you use tables. The limit as x will go in the middle box and then from the leftb and right the numbers get closer and closer then on the bottom you graph the function and hit trace on those values.
