1. What is continuity? What is discontinuity?
Continuity means the graph is
predictable; it has no breaks, holes, or jumps. It can be drawn without lifting
the pencil off the paper.
Discontinuity is when the graph can
contain breaks, holes, and jumps. It's separated into two families: Removable
Discontinuities and Non-Removable Discontinuities. Point discontinuity is
a removable discontinuity, meaning there's a hole in the graph. Non-Removable
Discontinuities has Jump Discontinuity, Oscillating Behavior, and Infinite
Discontinuity. Jump discontinuity means they have different left and right. You
must remember that the jump can have two open circles, one close/open, but they
must never be two closed circles. Oscillating behavior means the graph is very
wiggly, or unpredictable. The limit DNE because you can't really pinpoint the
graph's next destination. Infinite discontinuity is when there's a vertical
asymptote leading to unbounded behavior. Unbounded behavior means it increases
or decreases without bond. This is because the graph is either going up infinitely
or down infinitely.
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| Jump Discontinuity |
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| Point Discontinuity |
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| Oscillating Behavior |
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| Infinite Discontinuity |
A limit is the INTENDED height of a
function; whereas a value is the ACTUAL height of a function. A limit exists
when the intended and actual height are the same. A limit does not exist when
the left and right are different, unbounded behavior, and oscillating
behavior.
3. How do we evaluate limits
numerically, graphically, and algebraically?
Numerically means you use the
table. As the table goes towards the center, it's getting closer to the limit,
or y-value. It also helps us find if the limit can be reached or not.
Graphically is when we plug the function onto our calculator to graph and then
trace to the value you are looking for. To evaluate the limit, you put your
finger on a spot to the right and left; where your fingers meet is where the
limit is. If your fingers does not meet, then it does not exist. For
algebraically, there are three methods: Substitution, Factoring, and Conjugate.
Substitution is simply just plugging in the number into the function. You
always want to try this method first before doing any other method. If you get
0/0, that means it's an indeterminate form and you have to use another method.
The factoring method is when you factor the numerator and denominator and
cancel like terms to remove the zero. Then with the simplified expression, use
substitution to find your answer. The rationalizing/conjugate method is when you
multiply the radical with its conjugate. Remember to not multiply
the non-conjugate because something would hopefully cancel. Once something does
cancel, you use substitution to get your answer.
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| Numerically |
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| Algebraically |
Work Cited:
http://www.mathsisfun.com/calculus/continuity.html
http://en.wikipedia.org/wiki/File:Upper_semi.svg
http://www.wyzant.com/resources/lessons/math/calculus/limits/continuity
http://www.cwladis.com/math301/limitsgraphically.php
http://farmercalculus.wordpress.com/category/limits/
