Post on 11-Oct-2020
11.4part2Killeen.notebook
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April 26, 2019
Continuity and One Sided Limits
(special case of when lim L ≠ lim R)*finger test*
we want more information of where left and where right
the limit still DNE)
Objective: You will be able to:• define continuity• evaluate one sided limits• graph a piecewise function to represent the given information for limits.
Not in our book!
11.4part2Killeen.notebook
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April 26, 2019
A function is continuous at c if...
1. f (c) is defined (the point exists)
2. lim f (x) exists (the limit exists)
3. f (c) = lim f (x) (the point = the limit)
x c
x c
11.4part2Killeen.notebook
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April 26, 2019
Three conditions exist for which the graph of f is not continuous at x = c.
1. Why is this not continuous?
not continuous because f(c) is not defined(the point DNE)
11.4part2Killeen.notebook
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April 26, 2019
2. Why is this not continuous?
not continuous because lim f(x) DNEx c
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April 26, 2019
3. Why is this not continuous?
not continuous because f(c) ≠ lim f(x)
point DNE the limit
x c
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April 26, 2019
Discontinuities can be ...
1. Removable
2. Nonremovable
Only discontinuities at one spot and could be fixed by redefining f (c)
Big gaps or asymptotes, cannot be "fixed"
11.4part2Killeen.notebook
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April 26, 2019
1. Removable: only discontinuities at one spot and could be fixed by redefining f (c)
ex. f(x) = x2 ‐ 4
x ‐ 2
if f(x) = { x2 ‐ 4 x ‐ 2
4
, x ≠ 2
, x = 2
then it would becontinuous
f(x)= x+2twin function looks the same in all but one point
Rewrite as a piecewise function
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April 26, 2019
2. Nonremovable
Big gaps or asymptotes, cannot be "fixed"
ex. f(x) = 1
x
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April 26, 2019
Is it continuous or discontinuity?
Removable or Nonremovable?
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April 26, 2019
Classifying Discontinuities
Removable or point (Holes) 2 sided limit exists
Essential or Nonremovable
Jump1 sided limits exist
Infinite (vertical asymptotes) at least one of the 1 sided limits don't exist
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April 26, 2019
What is the difference between continuous and discontinuity?
What is the difference bewteen removable and nonremovable if it is a discontinuity?
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April 26, 2019
Find the constant "c" such that the function is continuous on the entire real line. Graph your answer and state what c must be.
f(x) = x + 1, x > 2 cx 4, x < 2
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April 26, 2019
GraphsSketch the graph on one function that satisfies all of the stated conditions.
Partner paper to be handed in. Must be on graph paper.
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April 26, 2019
GraphsSketch the graph on one function that satisfies all of the stated conditions.
ans
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April 26, 2019
GraphsSketch the graph of one function that satisfies all of the stated conditions.
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April 26, 2019
GraphsSketch the graph of one function that satisfies all of the stated conditions.
ans
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April 26, 2019
GraphsSketch the graph of one function that satisfies all of the stated conditions.
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April 26, 2019
GraphsSketch the graph of one function that satisfies all of the stated conditions.
ans
11.4part2Killeen.notebook
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April 26, 2019
ex. Find a and b so that f(x) is everywhere continuous.
Hint: Start here
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April 26, 2019