1. What is continuity? What is discontinuity?(a) A continuous function has no breaks, no holes, and no jumps in the graph. When drawing a continuous function you know it’s continuous if you draw it without lifting your pencil from the paper.
(b) A discontinuity is the opposite of continuity where
there can be breaks and holes in the graph.
2. What is a limit? When does a limit exit? When does a limit
not exist? What is the difference between a limit and a value?
(a) A limit is the intended height of a function.
(b) A limit can exist as long as you reach the same height
from both the left and right direction. Even if you were to have a hole you
will still have a limit because it is the intended
height, not the actual height.
(c) A limit does not exist at the 3 non-removable
discontinuities; when it has a jump, is unbounded behavior, or when it is oscillating
behavior.
(d) The difference between a limit and a value is that the
limit is the intended height of a function while the value of the limit can be
reached.
3. How do we evaluate limits numerically, graphically, and
algebraically?
(a) We evaluate limits numerically by using a table. What
we do is have x be number values that are approaching the intended height and
we will have f(x) be the number we get when plugged in and we find out if the
limit can be reached or not.
(b) We evaluate limits graphically by either using the calculator
or the picture given of the graph. When you see the two points from both sides
they need to meet up at the same point if they don’t then the limit doesn’t
exist.
(c) Lastly, we evaluate limits algebraically with direct substitution,
dividing out/factoring method, and rationalizing/conjugate method.
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