Continuity is predictable and it has no breaks, no holes and no jumps. A continuity function can be drawn with a single, unbroken pencil stroke for example the limit as x approaches a number of f(x) is equal to a number, the intended height.
What is discontinuity?
Discontinuity is the opposite of continuity meaning it's interrupted and the limit does not exist. There are two families of discontinuity, removable and non removable. Removable has point discontinuity meaning there's a hole in the graph. Non removable has jump-it jumps from one graph the other-,oscillating-the graph is wiggly meaning it never reaches a point- and infinite-when it has vertical asymptotes due to unbounded behavior.
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 when a value is the actual height of a function. The limit exist when the left and right behavior are the same, when there's no unbounded behavior and when there's no oscillating behavior. A limit doesn't when the left and right bahavior are different, there's an oscillating behavior and there's unbounded behavior.
How do we evaluate limits numerically, graphically, and algebraically?
We evaluate limits numerically, verbally and algebraclly. Numerically means its set on a table to find the intended the height of a function. Verbally is stateting the limit statement as the limit as x approaches a number of f(x) is equal to a number. Algrebraclly has three methods substitution, dividing & factoring out and rationaling & conjugate.
Direct substitution is when it's substituted by approaching number and has three answers numerical-2-, zero-0/2-, and undefined-1/0.
When its inderteinate meaning its 0/0 then the dividing & factoring method is use. In this case we factor out both the numerator and denominator to cancel out common terms. Then substitute and get either numerical-2-, zero-0/2-, or undefined-1/0.
In rationaling & conjugate is when we conjugate the numerator or denominator depending where the square root is at. After common terms are cancel then we substitute to get the answer.
Work cited
SSS packet
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