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§ 2-1 Limits
The student will learn about:limits,
infinite limits,
and uses for limits.
limits, finding limits, one-sided limits,
properties of limits,
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Procedure
During the next few days we will be studying limits. Limits form the foundation of calculus.
We will begin by studying limits from a graphical viewpoint.
We will end with the importance of limits.
We will then look at the properties of limits.
We will move on to the algebraic techniques needed to determine limits.
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Functions and GraphsA Brief Review
The graph of a function is the graph of the set of all ordered pairs that satisfy the function. As an example the following graph and table represent the same function f (x) = 2x – 1.
x f(x)
-2 -5
-1 -3
0 -1
1 1
2 ?
3 5
We will use this point on the next slide.
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Limits(THIS IS IMPORTANT)
Analyzing a limit. We can also examine what occurs at a particular point. Using the above function, f (x) = 2x – 1, let’s examine what happens when x = 2 through the following table:
x 1.5 1.9 1.99 1.999 2 2.001 2.01 2.1 2.5
f (x) 2 4
2
3
2.8 3.2 2.98 3.02 2.998 3.002?
Note; As x approaches 2, f (x) approaches 3. This is a dynamic situation
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Limits IMPORTANT!This table shows what f (x) is doing as x approaches 2. Or we have the limit of the function as x approaches We write this procedure with the following notation.
x 1.5 1.9 1.99 1.999 2 2.001 2.01 2.1 2.5
f (x) 2 2.8 2.98 2.998 ? 3.002 3.02 3.2 4
31x2lim2x
Def: We writeL)x(flim
cx
if the functional value of f (x) is close to the single real number L whenever x is close to, but not equal to, c. (on either side of c).
or as x → c, then f (x) → L 2
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H
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One-Sided Limit
This idea introduces the idea of one-sided limits. We write
and call K the limit from the left (or left-hand limit) if f (x) is close to K whenever x is close to c, but to the left of c on the real number line.
K)x(flimcx
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One-Sided Limit
We write
and call L the limit from the right (or right-hand limit) if f (x) is close to L whenever x is close to c, but to the right of c on the real number line.
L)x(flimcx
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The Limit
K)x(flimcx
Thus we have a left-sided limit:
L)x(flimcx
And a right-sided limit:
And in order for a limit to exist, the limit from the left and the limit from the right must exist and be equal.
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Example 1 f (x) = |x|/x at x = 0
1x
xlim
0x
The left and right limits are different, therefore there is no limit.
01x
xlim
0x
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Example 2
Example from my graphing calculator.
4)x(flim4x
The limit does not exist at 2!
4)x(flim4x
4)x(flim4x
The limit exists at 4!
2 42
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4)x(flim2x
2)x(flim2x
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Infinite Limits
Sometimes as x approaches c, f (x) approaches infinity or negative infinity.
2
1f (x)
x 2
Consider
Notice as x approaches 2 what happens to the y values.
2x 2
1lim
x 2
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Infinite Limits
Be careful.
1
f (x)x 2
Consider
Notice as x approaches 2 what happens to the y values.
x 2
1lim does not exist!
x 2
The left limit is - ∞ while the right limit is + ∞.
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Limit Properties• the limit of a constant is just the constant.
aalimcx
• the limit of a power is the power of the limit.
nn
cxcxlim
• the limit of the root of the root of the limit.
nn
cxcxlim
(If n is even then c ≥ 0 )
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Limit Properties Continued
Let f and g be two functions, and assume that the following two limits are real and exist.
• the limit of the sum or difference of the functions is equal to the sum of the limits.
Then:
M)x(glimandL)x(flimcxcx
ML)x(glim)x(flim)x(g)x(flimcxcxcx
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Limit Properties Continued
If:
Then:
M)x(glimandL)x(flimcxcx
• the limit of the product of the functions is the product of the limits of the functions.
ML)x(glim)x(flim)x(g)x(flimcxcxcx
• the limit of the quotient of the functions is the quotient of the limits of the functions.
0)x(glimif
M
L
)x(glim
)x(flim
)x(g
)x(flim
cxcx
cx
cx
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Summary of Rules of Limits
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Example 1
x2 – 3x = 2x
lim 2 2 – 3 · 2 = 4 – 6 = -2.
Many times we may find a limit by direct substitution.
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Summary.
• We learned about left and right limits.
• We learned the definition of limit.
• We learned about limits of infinity.
• We learned about properties of limits.
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Summary from Text.
• If a function exist to the left and right of a given value the left and right limits will usually exist. Remember at a vertical asymptote the lift and right limits will be ± ∞.
• The graph can help us roughly determine limits.
• When left and right limits are not equal the limit does not exist.• At vertical asymptotes left and right limits may be equal, hence a limit exist, or them may not be equal and then there is no limit.
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ASSIGNMENT
§2.1; Page 36; 1 – 5.
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Algebraic LimitsAlthough graphs are often very useful in finding limits there are algebraic methods that are quick and accurate.
Is one of those situations that comes to mind.x a
lim f (x) f (a)
However one must be careful in the case where
x a
0lim f (x) .
0
In this situation algebraic techniques to remove the part of the denominator causing the problem are necessary
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Indeterminate Form
The term indeterminate is used because the limit may or may not exist.
if and , then
are said to be indeterminate.
x alim f (x) 0
x alim g(x) 0
x a
f (x) 0lim
g(x) 0
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If orx a
0lim f (x)
0
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Example 3
Be careful when a quotient is involved.
!undefinedisWhich0
0
2x
6xxlim
2
2x
2
x 2 x 2
x x 6 (x 3)(x 2)lim lim
x 2 x 2
However
?Why.linestraightaasgraphs2x
6xx)x(f:NOTE
2
What happens at x = 2?
x 2lim (x 3) 5
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Example 4
x 4
4 x 0lim Which is undefined!
02 x
x 4 x 4
4 x (4 x)(2 x)lim lim
2 x (2 x)(2 x)
Actually, notice it does factor but this was so much more fun!
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Notice that this one does not factor but we can be algebraically creative!
x 4
(4 x)(2 x)lim
4 x
x 4lim 2 x 4
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Importance of Limits.
• Limits will be the building block of both of our major definitions in calculus and calculus has such very wide applications.
• The idea of a limit has always intrigued mathematicians. The idea of getting closer and closer to something and never reaching it has a certain Zen quality.
• Indeed you will find that the type of limit we always need to determine will be a 0/0 limit.
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Commentary
This course is devoted to the development of
the important concepts of the calculus,
concepts that are so far reaching and that
have exercised such an impact on the
modern world that it is perhaps correct to
say that without knowledge of them a
person today can scarcely claim to be well
educated. Howard Eves
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Summary.
• We learned that the indeterminate limit the 0/0 limit will become important to us.
• We learned about algebraic method for finding limits.
by Tony Carrillo
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ASSIGNMENT
§2.2; Page 36; 1 – 10.
