Lesson 4 Limits
Announcements
Popper 2 is today
Homework
Quiz
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Definition
We say that a function f has a limit L as x approaches the target number a
if the value of f(x) can be made as close to L as we please by taking x sufficiently
close to (but not equal to) a. Note that L is a single real number!
Well, we need to tackle several things here. First, what does this actually mean?
Next, how do you say it in “manglish”?
Third where does it show up on graphs and problems?
Let’s look first at a graph of a piecewise defined function:
Let’s look at the limit as x approaches 0 and the limit as x approaches – 2.
Now here’s some news – L, the limit, is a y-value!
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Popper 02 Question 1
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So now let’s “approach” x = 0…note there’s a left hand approach and a right hand
approach. What are the y’s as we do this?
Note that the y’s change as we move through the x’s…
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Now let’s approach −2 what y does it look like we’re heading for?
Now −2 is known as our “target” and
Approach −2 from the left and the right….where are the associated y values going
as we move?
BTW – the graph is not from GGB; it’s from Graph! and that program is a free
download, too.
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Now, you don’t imagine that I say or write ALL those words when I need a limit, I
hope. There’s a much more abbreviated way to say it:
4 is the limit of f ( x) as x approaches −2 :
lim f ( x)  4
x 2
MUCH shorter, but you have to be able to translate it.
Popper 02
Question 2
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Another example:
Given the following graph,
lim f ( x)  _____
x 1
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Now sometimes a limit does not exist (DNE). Let’s look at a couple of those.
Example of DNE with a vertical asymptote:
lim f ( x) 
x 0
The approach from the left and the right do not go to a number at all.
However:
lim (1/ x) 
x 1
You can have them and not have them point by point on the same graph!
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Popper 02 Question 3
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Example:
The approach from the left and right do not go to the same number. I’ll use a step
function to illustrate this but there are many piecewise functions that exhibit this
behavior. Where do step functions show up in real life?
lim f ( x) 
x2
However:
lim f ( x) 
x 1
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Now, it’s not always the case that a function has a value at the target point.
Remember back on the first page when I wrote
We say that a function f has a limit L as x approaches the target number a
if the value of f(x) can be made as close to L as we please by taking x sufficiently
close to (but not equal to) a.
Let’s look at this example:
This is a piecewise defined function that is NOT defined at f (0), the y-intercept.
f ( x)
BUT let’s look at lim
x 0
Do you see that limits let you work where functions are not defined because we
don’t have to actually USE the target value?
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Some nice things to know about limits, known as Properties of Limits:
Suppose the lim
f ( x)  P and lim g ( x)  Q  0 , suppose c is a real number.
x a
x a
lim cf ( x)  cP
xa
lim[ f ( x)  g ( x )]  P  Q
xa
lim[ f ( x) g ( x)]  PQ
xa
lim[
x a
f ( x)
P
]
g ( x) Q
Popper 02
Question 4
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Problems
Suppose f ( x)  x2  4
Evaluate:
lim f ( x) 
x2
Popper 02 Question 5
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Suppose f ( x) 
Popper 02
3x  1
x 9
Evaluate:
lim f ( x )
Evaluate:
lim f ( x )
x 9
x2
Question 6
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Suppose f ( x)  7 x
2
3
Evaluate:
lim f ( x )
x 8
Popper 02 Question 7
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By now you must be wondering about where GGB fits in. Well, when you’re
working along and substitution gives you a function with a vertical asymptote and
you are supposed to evaluate a limit at that asymptote…you may also use GGB.
If you are looking at at something like
k
, this is called an indeterminate form.
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Command: limit[<Function>, <Value>]
It is advisable to use GGB if you really can’t visualize the graph. Here’s a sneaky
one:
f ( x) 
x 2
x4
lim f ( x )
x4
This is not your usual line/line graph. That square root puts in some glitches.
Let’s look at the graph and then I’ll show you with algebra what is going on.
Where is the asymptote?
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If you take the limit with GGB you get ¼.
Let’s look hard at this function.
Note ( x )2  x
( x  2)( x  2)  x  4
x 2

x4
1
x 2
by cancelling
If you can’t imagine the graph, do it in GGB
If you don’t have an indeterminate form (something/0) you may just evaluate the
function at the limit.
Popper 02 Question 8
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Sometimes the target is “infinity”…let’s look at some situations where we want to
take a limit at infinity…we say this
lim f ( x)
x 
And there are only a few things that can happen.
IF there is a horizontal asymptote then the limit is the HA number.
If there is not, then the limit is infinity or negative infinity.
Reviewing
HA
Check the ratio of the exponents of the leading coefficients:
N=D
N>D
N<D
HA is the ratio of the leading coefficients
no HA
HA is the x-axis, called y = 0
Leading term analysis of the end behavior
f ( x)  ax n  junk
a is positive
n is even
n is odd
a is negative
n is even
n is odd
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So let’s check on some:
f ( x) 
9 x 7  11x
3x 7  x5
f ( x)   x11  5x3  2
f  x 
x3
x 2  200
5 x3
f ( x) 
x 1
lim f ( x) 
x 
lim f ( x) 
x 
lim f ( x) 
x 
lim f ( x) 
x 
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Popper 02 Question 9
Popper 02 Question 10
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