Continuity, Discontinuity, Limits, and How to Evaluate Them
1.) Continuity is being able to draw a function without lifting your pencil off the paper. There are no breaks, jumps, or holes in a continuous function. It is also predictable, the function goes where you think it should go, this means that the limit (intended height) will be the same as the value (the actual height). For example, if the limf(x) x->3= 5 and f(x)=5, then that function is continuous. A continuous function can be thought of as a bridge, you can't fall through (no hole/point discontinuity) and you can't fall off (no jump discontinuity. A discontinuity can be categorized into two sections, removable and non-removal. They are separated because the limit only exists at removable and the limit DNE at non-removable. The only removable discontiinuity is called a point discontinuity and it is represented by a hole. In the non-removable category there are jump dicontinuity, infinite discontinuity, and oscillating behavior. In a jump discontinuity, the limit DNE exist because of different left and right limits. If the limits are different on opposite sides, then it is unclear what the intended height is. Similarly, in an infinite discontinuity an asymptote causes the graph to go up or down in infinite directions, thus resulting in a limit that DNE due to unbounded behavior. Oscillating behavior results in a limit that DNE because the function does not approach any single value.
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| Types of Discontinuities: https://docs.google.com/viewer?a=v&pid=sites&srcid=ZGVmYXVsdGRvbWFpbnxkb2NzZm9ya2lyY2hzY2xhc3Nlc3xneDoxYWY2ZjAyMWIwMTI5YWJj |
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| Here the limit is the same as the value, thus the function is continuous. - https://docs.google.com/viewer?a=v&pid=sites&srcid=ZGVmYXVsdGRvbWFpbnxkb2NzZm9ya2lyY2hzY2xhc3Nlc3xneDoxYWY2ZjAyMWIwMTI5YWJj |
2.) A limit is the intended height of a function (Mrs. Kirch). It is the y-value that is reached the nearest when approaching a specific x-value of a function. Limits exist at holes, which is why point discontinuity has limits. this is because the limit is not the actual height, but the intended height. A limit exists when you reach the same height from both the left and right hand sides. In class we have been using the analogy of driving to a diner to evalute limits graphically. If your fingers meet at the same spot from left and right directions, then the limit exists. To evaluate this algebraically, we use left and right hand side limits and denote them with + for right and - for left. If these limits are the same, a limit exists. A limit does not exist if the values in the left and right hand limits are different. A limit is the intended height of a function whereas the value is the actual height of a function. When the limit and the value are the same we have a continuous function.
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| An example of a graphical expression but also different types of limits. The value of a function is undefined at a hole but the limit still exists. The value would be wherever the hole is shaded. This is unless it is a jump discontinuity, as seen in the image. - https://docs.google.com/viewer?a=v&pid=sites&srcid=ZGVmYXVsdGRvbWFpbnxkb2NzZm9ya2lyY2hzY2xhc3Nlc3xneDoxYWY2ZjAyMWIwMTI5YWJj |
3.) Numerically, we evaluate limits by using a t-table with approximate x-values and use a graphing calculator to find the corresponding y-values. Whatever value is reached the closest would be our limit. Graphically, we use our fingers and approach a certain value from the left and right hand sides. If our fingers meet, then the limit is at that height. If they do but the circle is not shaded, then it is a hole and the limit still exists. Similar to our diner analogy, this means the diner is burned down but we still reached the destination, a.k.a. the intended height. Algebraically we use three methods to evaluate limits: direct substitution, dividing out, and rationalizing. In direct substitution, we plug in the exact x-value and solve. When that results in 0/0 thenw e must resort to the dividing out method in which we look to factor numerator and denominator and hope something cancels so we can then substitute directly. If nothing factors, then use the rationalizing method; in this method, we use a conjugate to cancel out a part of a function in order to solve for the rest. What results would be the value of our limit.
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| Evaluating a function numerically, here the limit would be 6, as it is the nearest approached value. - https://docs.google.com/viewer?a=v&pid=sites&srcid=ZGVmYXVsdGRvbWFpbnxkb2NzZm9ya2lyY2hzY2xhc3Nlc3xneDoxYWY2ZjAyMWIwMTI5YWJj |
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| This algebraic solution shows how using a conjugate can help cancel out an x-value. That followed by direct substitution gives you the limit. - https://docs.google.com/viewer?a=v&pid=sites&srcid=ZGVmYXVsdGRvbWFpbnxkb2NzZm9ya2lyY2hzY2xhc3Nlc3xneDoxYWY2ZjAyMWIwMTI5YWJj |
REFERENCES:
Mrs. Kirch's SSS packet - https://docs.google.com/viewer?a=v&pid=sites&srcid=ZGVmYXVsdGRvbWFpbnxkb2NzZm9ya2lyY2hzY2xhc3Nlc3xneDoxYWY2ZjAyMWIwMTI5YWJj
