Download Popper 03 Question 5

Lesson 5:
One-sided Limits and Continuity
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Popper 03 today
Quiz
Homework
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We’ve seen a step function before. Let’s look at one again to discuss one-sided
limits. First a bit of notation:
Let’s discuss the x-axis with 1 as our target number. We have all the numbers on
the left, less than 1 and we have all the numbers on the right, greater than 1. We
denote the numbers on the left with 1 and those on the right with 1 .
And we use this with the limit notation in the subscript about approaching the
target number:
lim f ( x)
x 1
means to approach 1 through values slightly less than 1
while
lim f ( x)
x 1
means to approach 1 through values slightly more than 1
Popper 03 Question 1
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Let’s work with this step function:
First
f (1) = _____
Now the limits:
lim f ( x)
x 1
lim f ( x)
x 1
Now for THE limit as opposed to the one-sided limits:
lim f ( x )
x 1
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Sometimes you only want to discuss information on one side of a limit or the other
and you have one-sided limits for that. Sometimes you need to know that there is a
“jump” in the graph there.
Theorem
Let f be a function that is defined for all values of x close to the target number a,
except perhaps at a itself.
Then the lim
f ( x ) = L if and only if lim f ( x)  lim f ( x)  L .
xa
x a 
x a 
So, if the one-sided limits agree, you’re good.
Popper 02 Question 2
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Let’s look at a piecewise defined function and talk about the limits.
 x2  x  2

f ( x)   x  1
 x2  4

x 1
1 x  2
x2
What about a limit at −1? And now at these x-values:
1
2
3
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Popper 03 Question 3
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Continuity
We want to discuss the continuity of a given function at a point, over an interval,
and for the whole domain. So first let’s get a definition of continuity. Briefly, a
function is continuous at a point if the graph has no gaps, breaks, or jumps at that
point.
More formally,
A function f is said to be continuous at a point x = a if the following three
conditions are met:
f (a) is defined,
the limits exists, and
the value and the limit are the same number.
If a function cannot meet these criteria, then we say it is discontinuous at a.
Discontinuities come in several flavors: jump, removable, and infinite are the ones
we’ll explore.
Jump discontinuities happen when the one-sided limits are not equal.
If there is a hole where the limit should be, it is called a removable discontinuity.
Infinite discontinuities happen at vertical asymptotes when the graph is heading off
to infinity or negative infinity.
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Popper 03 Question 4
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Here’s one with just about everything. Let’s go from left to right:
See the VA at −2? Let’s look at which function is there. Let’s write out the
piecewise defined function from the graph:
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So, reviewing, we’ve got
Jump discontinuities
Popper 03 Question 5
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Removable discontinuities
Popper 03 Question 6
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Infinite discontinuities
Popper 03 Question 7
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Now let’s work without a graph for a moment:
x  6
f ( x)   2
 x  5x  6
x0
x0
Can we figure out what’s going on at zero WITHOUT a graph? Is the graph
continuous at zero? How will we check?
What is the domain for this function? How do we figure out limits everywhere
BUT zero? Are there ANY possibilities for a discontinuity anywhere else?
What are we looking for when we’re searching for discontinuities?
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Popper 03 Question 8
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Another example:
f ( x) 
Is this function continuous at x = −5?
x 2  25
5 x
What’s the domain? What’s the issue right at x = −5? Let’s work with the algebra
a bit!
What does the graph look like? From Graph!
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Did the technology let you down a bit?
What kind of problem is this? Let’s look at how to fix it!
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Another example:
f ( x) 
2x  2
x 1
What is the domain?
Where are the asymptotes?
What about limits?
At x = 1, x approaches negative infinity?
What about discontinuities?
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Popper 03 Question 9
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Continuity over an interval
A function is continuous over an open interval (a, b) if it is continuous at every
point in the interval…note that the endpoints are not included.
Where is the function above continuous?
Where is f ( x)  x4  5x 2  2 x  7 continuous?
Where is f ( x) 
7x
continuous?
x  5 x  17
2
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An application of limits:
The average cost in dollars of constructing each skateboard when x = the number
of skateboards constructed can be modeled by the function
C ( x)  12.5 
123,500
x
What is the average cost per skateboard if the number produced gets larger and
larger?
Popper 03 Question 10
Vicious and Cruel…read carefully!
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