A continuity is a graph that has no jumps, no breaks, and no holes. The graph itself is predictable so you can draw it without lifting your pencil off the paper. A discontinuity has two families which include a removable discontinuity and non-removable discontinuities . Removable discontinuity is consistent of the point discontinuity. This discontinuity has a hole. Non-removable discontinuities include jump. oscillating, and infinite discontinuity. Jump discontinuities have different left and right because they never meet up. Oscilatting is a wiggly line. Infinite discontinuity has unbounded behavior and vertical asymptotes.
#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?
The limit is the intended height of a function. A limit exists on a continuous graph or the removable discontinuity. With the left and right going towards the limit, it is able to exist. The limit doesn't exist at non-removable continuities, which are stated above as jump, oscilatting and infinite. A limit doesn't exist on jump because of different left and right, they never meet up so there is no limit. On oscillating theres no single point because its wiggly and keeps on moving. A limit doesn't exist as infinite discontinuity because of unbounded behavior and its vertical asymptote.
The limit is the intended height of a function while the value is the actual height of a function. Taking the first graph as an example there is a limit ,but theres no value since theres a hole. In some cases, for example the graph below the first one there will be a different limit and a different value somewhere else. The jump discontinuity has no limit because it jumps, it has no meeting place, however the value still exists at the closed circle. Oscillating and infinite discontinuity are never at an exact point.
#3: How do we evaluate limits numerically, graphically, and algebraically?
To evaluate limits graphically you can create a table, you choose your base number, then add and subtract by a tenth. So for example we can take the number 5 if we subtract a tenth from 5 it becomes 4.9. Once we add a tenth to 5 it becomes 5.1. Basically you want to get as close to your base number of 5 So to get closer from 4.9 would be 4.99 and 4.999. Closer to 5.1 would be 5.01, and 5.001.
To evaluate graphically we take our fingers from left and right and work our way towards the center.
Algebraically we used direct substitution in which we plug it in and solve, If that doesnt work we go to the factoring method in which we factor out the numerator and denominator and cancel terms to remove the zero in the denominator. Then we can use the rationalizing method in which we change the sign and multiply by our square root problems.
Reference: SSS Packet
To evaluate limits graphically you can create a table, you choose your base number, then add and subtract by a tenth. So for example we can take the number 5 if we subtract a tenth from 5 it becomes 4.9. Once we add a tenth to 5 it becomes 5.1. Basically you want to get as close to your base number of 5 So to get closer from 4.9 would be 4.99 and 4.999. Closer to 5.1 would be 5.01, and 5.001.
To evaluate graphically we take our fingers from left and right and work our way towards the center.
Algebraically we used direct substitution in which we plug it in and solve, If that doesnt work we go to the factoring method in which we factor out the numerator and denominator and cancel terms to remove the zero in the denominator. Then we can use the rationalizing method in which we change the sign and multiply by our square root problems.
Reference: SSS Packet
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