A continuous function is predictable (goes where you think it should go). It also has no holes, no jumps, and it can be drawn without lifting your pencil/no breaks in the graph. A discontinuity is made up of two families, which consist of removable discontinuities and non-removable discontinuities. A removable discontinuity has a point of discontinuity, which is also known as having a hole. A non-removable discontinuity has jumps, is unpredictable, you lift your pencil when you draw it, has vertical asymptotes, has different limits from values. A jump discontinuity has the right and left different. An oscillating behavior is wiggly. An infinite discontinuity, which is also known as unbounded behavior, occurs where there is vertical asymptote, which increases or decreases without bound of infinity or negative infinity.
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 a function. A limit exists at a point of discontinuity, which is also known as a removable discontinuity or hole. A limit exists as long as you reach the same height from both the left and right. In order for a limit to exist, both the right hand limit and left hand limit must be the same, so the limit does not exist at a jump discontinuity (left&right different), oscillating (doesn't have a certain point/wiggly), and infinite discontinuity (has a vertical asymptote). A limit is the intended height, but a value is the actual height.
3. How do we evaluate limits numerically, graphically, and algebraically?
Numerically: You set up a table to help evaluate limits. You first draw a table and you add 1/10 on one side, and subtract 1/10 to the other side; with this, you slower get closer to the middle number.
Graphically: To help evaluate limits, you can draw a graph to help you pain a picture to where your limits are and it helps you explains when why you get undefined for your values. Or, you can use your fingers to help you. You start on one side and the other finger on the other side, and you try to make them come together to see if it's a removable or non-removable discontinuity.
Algebraically: To help evaluate limits, you do problem solving by using three methods: direct substitution method, factoring method, and conjugate method. When you use the direct substitution method, all you do is plug in the number that x approaches and you're done. But if you get 0/0 or indeterminate form, you keep on going to the other two methods. When you use the factoring method, you factor the denominator and numerator, and you then cancel what you can, but remember to not combine when you can cancel something from it, but remember, to always try direct substitution first! When using the conjugate method, you multiply to where the radical is and multiply by "one" and after you cancel, you get your answer.
References:
Mrs. Kirch's Unit U SSS Packet
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