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2.2 Limits. Definition of Limit: Suppose that f(x) becomes arbitrarily close to the number L ( f(x) L ) as x approaches a ( x a ). Then we say that the limit of f(x) as x approaches a is L and write. Note 1: The number a may be replaced by ∞. 1. Example 1: - PowerPoint PPT Presentation
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2.2 Limits
Note 1: The number a may be replaced by ∞
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Definition of Limit: Suppose that f(x) becomes arbitrarily close to the number L ( f(x)L ) as x approaches a ( xa ).Then we say that the limit of f(x) as x approaches a is L and write
L=xfax
lim
2
Example 1: Estimate the limit of the function
as x approaches 2.
22 +xx=xf
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Example 2: Find the limit
Note 2: In general, the limit has nothing to do with the value of the function at a. All we are claiming is that the function approaches L.
In the previous example 42lim2
=f=xfx
1
1lim
21
x
x
x
Solution: The function under the limit is not defined at x=1, but this doesn't matter because the definition of limit says that we consider value of x that are close to a, but not equal to a.
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Provided that x ≠ a, the function can be reduced as follows:
1
1
11
1
1
12 +x
=+xx
x=
x
x
Then0.5
11
1
1
1lim
1
1lim
12
1
=+
=+x
=x
x
xx
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If the value of the limit coincides with the value of the function, the function is called continuous at this point.
Definition of Continuity: A function f is continuous at x=a if f is defined at a and
af=xfax
lim
Exercises: Determine if the functions in the above examples are continuous at the given points (points of limits)
2at ;22 xxx
1at ;1
12
xx
x
1at ;1
1
x
x
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Example: Find the trigonometric limit
x
x
x
)sin(lim
0
1)sin(
lim0
x
x
xContinuous at 0?
One-sided limits
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Definitions: If f(x) becomes arbitrarily close to the number L as x approaches a from the left, then we say that L is the left-sided limit of f(x) as x approaches a and write
If f(x) becomes arbitrarily close to the number L as x approaches a from the right, then we say that L is the right-sided limit of f(x) as x approaches a and write
Lxfax
lim
Lxfax
lim
Example: Heaviside function
8
0 1
0 ,0)(
x,
xxH
0 lim0
xHx
1 lim0
xHx
Continuous at 0?
Exercise 3: Find the limit
9
43
132lim 2
2
x
xx
x
Solution: We use the fact that 1/x approaches 0 as x increases (or approaches infinity). So, we divide both numerator and denominator by the highest power of x and then use the above limit of 1/x.
3/2
...43
132lim 2
2
x
xx
x
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Theorem: If and then
A.
B.
C.
D.
L=xfax
lim
M=xgax
lim
ML=xgxfax
)(lim
LM=xgxfax
)(lim
)0( )(
)(lim
M
M
L=
xg
xf
ax
kL=xkfax
)(lim
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Three types of limit calculation (see numbered exercises):1. Limit of a continuous function: just substitute the
number.2. 0/0 limit: cancel the common factor that gives 0 in the
numerator and denominator and then substitute the number.
3. ∞/∞ limit (ratio of polynomials at ∞): divide both numerator and denominator by the highest power of x and use the fact that the limit of the reciprocal function at ∞ is 0.
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HomeworkSection 2.2: 11,15,27,31,37,41,49,51.
