A continuous function is predictable. It has no holes, no breaks, and no jumps. It can be drawn with a single, unbroken pencil stroke. A continuous graph has to have the SAME limit and value; meaning the limit can only be reached with a continuous graph.
What is Discontinuity?
A discontinuity is any type of break, hole, or jump from a graph. The limit and the value of a discontinuous graph is always different from the value. There are two families of discontinuity: Removable and Non- Removable. Removable discontinuities (also known as holes), the limit does exist, but the value id undefined. Non- Removable discontinuities, the limit Does Not Exist, but the value can.
http://www.zweigmedia.com/RealWorld/calctopic1/pics3/pic1.gif
2. What is a limit?
A limit is the intended height of a graph. the limit can be a numerical, zero, undefined, and indeterminate. The limit DNE in Non-Removable discontinuities: jumps (comparing left and right behavior), oscillating, and infinite (unbounded behavior).
When does a limit exist?
A limit exist when a height is being shown with the exceptions of : two opened circles, and any open circle in general.
http://i45.tinypic.com/152hn36.jpg
When does a limit not exist?
A limit does not exist when we have opened circles, and wherever there may be vertical asymptotes. If the answer to a limit is 0/a number, the limit DNE. having different points on an X-point (jump) the limit DNE because of different Left and Right. Oscillating behavior, the limit DNE.
What is the difference between a limit and a value?
A limit is the INTENDED height of a graph, does not mean the value is the same.
A value is the ACTUAL height of a graph, it can be determined as undefined at times.
3. How do we evaluate graphs numerically, graphically, and algebraically?
Numerically is evaluated with a table where we get VERY, VERY, VERY, VERY, VERY, close to the point, but never reach it.
https://finitemathematics.wikispaces.hcpss.org/file/view/limit_table.PNG/239144381/575x194/limit_table.PNG
We evaluate Graphically by using two fingers and following our graph to see if they meet. The identification of the graph and its discontinuities, and continuities.
http://www.coolmath.com/precalculus-review-calculus-intro/precalculus-algebra/images/21-graphing-04.gif
We evaluate Algebraically by using the direct substitution, dividing out/ factoring method, and rationalizing/ conjugate method.
http://www.analyzemath.com/calculus/limits/find_limit_3_2.gif
