1.2­Limits graphically, numerically.notebook
September 12, 2013
Warm­Up­­Sept. 12, 2013
*This problem is very similar to #6 on the 1.1 homework.*
Secant Lines: Consider the function f(x) = 6x ­ x2 and the point P(2, 8) on the graph of f.
a) Graph f and the secant lines passing through P(2, 8) and Q(x, f(x)) for x­values of 3, 2.5, and 1.5.
b) Find the slope of each secant line.
c) Use the results of part (b) to estimate the slope of the tangent line to the graph of f at P(2, 8). Describe how to improve your approximation of the slope.
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1.2­Limits graphically, numerically.notebook
September 12, 2013
Warm­Up Solution
Each part is worth 3 points for a total of 9 points, just like the free response on the exam.
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1.2­Limits graphically, numerically.notebook
September 12, 2013
1.2 Finding Limits Graphically &
Numerically
To define a "limit," we will take a look at the function
The LIMIT of f(x) is the BEHAVIOR of the function but not
necessarily the VALUE of the function.
Let's look at the limit of the function as x gets close to 2:
To get an idea of the behavior of the graph of f near x = 2,
we can use two sets of x-values--one set that approaches 2
from the left and one set that approaches 2 from the right.
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1.2­Limits graphically, numerically.notebook
1.9
x=2
to side
e
s
t
Clo
lef
m
fro
1.99
3.71
3.9701
1.999 3.997001
September 12, 2013
2.1
2.01
4.31
4.0301
Clo
fr se t
om
o
rig x=2
ht
sid
e
2.001 4.003001
We can see the function has a limit =______ when x approaches 2
from the left and from the right.
We write it like this:
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1.2­Limits graphically, numerically.notebook
September 12, 2013
In general terms:
"the limit of f(x) as x approaches a is L"
• The limit of a function at a point is the value that the function approaches but never reaches. (It can equal the value but doesn't have to.)
• The existence or nonexistence of f(x) at x = c has no bearing on the existence of the limit of f(x) as x approaches c.
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1.2­Limits graphically, numerically.notebook
September 12, 2013
Graphs:
y=f(x)
y=f(x)
y=f(x)
a
f( )
L
L
L
and
a
a
a
but
since f(a) is not defined for y = f(x)
but
although f(a) is not defined for y = f(x)
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1.2­Limits graphically, numerically.notebook
September 12, 2013
Example:
Let's find the limit of
as x approaches 1.
We will look at the graph and a table.
The graph of f is a parabola that has a gap at the point (1, 3).
Although x can not equal 1, you can move arbitrarily close to 1, and as a result f(x) moves arbitrarily close to 3. Using limit notation, you can write________
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1.2­Limits graphically, numerically.notebook
September 12, 2013
Now, let's look at the function:
What is the domain?_____________
Graph it on the calculator & examine 1st quadrant only.
Since there's a "hole," let's give it a
value at x = 1 by redefining the function
using a piecewise function.
Window:
x­min=0
x­max=2
y­min=0
y­max=2
Does redefining it change its behavior around x = 1?
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1.2­Limits graphically, numerically.notebook
September 12, 2013
Let's use the calculator to help us find its behavior...
Type function into Y 1 and go to the TableSet.
Start at 0.90 and use ΔTbl = 0.05.
What value does it seem to be approaching?
Now, redefine it so that it flows as one continuous curve.
Therefore, even though f(1) is undefined (when not written as a piecewise).
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1.2­Limits graphically, numerically.notebook
September 12, 2013
Limits That Fail to Exist
"Stay on the path to the graph!"
If your fingers don't touch on both sides of the limit in question, then the limit does not exist. Behavior That Differs from the Right and from the Left
does not exist
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1.2­Limits graphically, numerically.notebook
September 12, 2013
Unbounded Behavior
does not exist
• f(x) increases without bound
• By choosing x close enough to 0, you can force f(x) to be as large as you want.
• Because f(x) is not approaching a real number L as x approaches 0, you can conclude that the limit does not exist.
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1.2­Limits graphically, numerically.notebook
September 12, 2013
Oscillating Behavior
As x approaches to 0 , sin(1/x) keeps oscillating near the
y -axis between 1 and -1 but it does not approach to
anywhere.
Therefore, does not exist.
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1.2­Limits graphically, numerically.notebook
September 12, 2013
Summary of Nonexistent Limits
1.
2.
3.
f(x) approaches a different number from the right side of c than it approaches from the left side
f(x) increases or decreases without bound as x approaches c
f(x) oscillates between two fixed values as x approaches c
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