5 2 limit-ofa_function
TRANSCRIPT
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5.2 Limit of a Function5.2 Limit of a Function
Limit of a Function
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5.2 Limit of a Function5.2 Limit of a Function
• The concept of a limit is central to the study of calculus.
• Differentiation and integration are each based on the notion of a limit.
• Limit of f (x) as x approaches a – the value that f (x) approaches as x gets closer to a
• In symbols:
€
limx → a
f (x)
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5.2 Limit of a Function5.2 Limit of a Function
Observe the values of f (x) = x2 + 1 as x approaches 3:
We say that .
€
limx → 3
f (x) =10
x 2 2.5 2.9 2.99 2.999 3 3.001 3.01 3.1 3.5 4
f (x) 5 7.25 9.41 9.94 9.994 10 10.006 10.06 10.61 13.25
17
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5.2 Limit of a Function5.2 Limit of a Function
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5.2 Limit of a Function5.2 Limit of a Function
Given . Find the limit of
f (x) as x approaches 3..
We still say that .
€
limx → 3
f (x) =10
x 2 2.5 2.9 2.99 2.999 3 3.001 3.01 3.1 3.5 4
f (x) 5 7.25 9.41 9.94 9.994 15 10.006 10.06 10.61 13.25
17
€
f (x) =x 2 +1 for x ≠ 3
15 for x = 3
⎧ ⎨ ⎩
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5.2 Limit of a Function5.2 Limit of a Function
Given . Find the limit of
f (x) as x approaches 2..
€
limx → 2−
f (x) = 5
x 1 1.5 1.9 1.99 1.999 2 2.001 2.01 2.1 2.5 3
f (x) 3 4 4.8 4.98 4.998 4 4.004 4.04 4.41 6.25 9
€
f (x) =2x +1 for x < 2
x 2 for x ≥ 2
⎧ ⎨ ⎩
€
limx → 2+
f (x) = 4
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5.2 Limit of a Function5.2 Limit of a Function
• The limit of the function f exists only if the left-hand and right-hand limits both exist and are equal.
• exists if and only if
€
limx → a
f (x)
€
limx → a −
f (x) = limx → a +
f (x) = L
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5.2 Limit of a Function5.2 Limit of a Function
Limit Theorems
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5.2 Limit of a Function5.2 Limit of a Function
L1. The limit of a constant function is the constant.
Examples.
€
limx → a
k = k
€
limx → 5
− 2 = −2
€
limx → −3
32 = 3
2
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5.2 Limit of a Function5.2 Limit of a Function
L2. If f is a polynomial function, then
Example.
€
limx → a
f (x) = f (a)
€
limx → 3
(x 3 + 6x) = 33 + 6(3) = 45
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5.2 Limit of a Function5.2 Limit of a Function
L3. The limit of a sum is the sum of limits.
Example.
€
limx → a
f (x) + g(x)[ ] = limx → a
f (x) + limx → a
g(x)
€
limx → 2
x 2 + 2x − 3( ) = limx → 2
x 2 + limx → 2
2x − limx → 2
3
= 4 + 4 − 3
= 5
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5.2 Limit of a Function5.2 Limit of a Function
L4. The limit of a constant times a function is the constant times the limit of the function.
Example.
€
limx → a
kg(x) = k limx → a
g(x)
€
limx → 5
12 (x + 3) = 1
2 limx → 5
(x + 3) = 12 (8) = 4
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5.2 Limit of a Function5.2 Limit of a Function
L5. The limit of a product is the product of the limits.
Example.
€
limx → a
f (x)⋅ g(x)[ ] = limx → a
f (x)[ ] limx → a
g(x)[ ]
€
limx →1
(x + 3)(x 2 +1) = limx →1
(x + 3)⋅ limx →1
(x 2 +1)
= 4⋅ 2
= 8
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5.2 Limit of a Function5.2 Limit of a Function
L6. The limit of a quotient is the quotient of the limits.
Example.
€
limx → a
f (x)
g(x)=
limx → a
f (x)
limx → a
g(x), lim
x → a g(x) ≠ 0
€
limx → 2
x − 3
x +1=
limx → 2
(x − 3)
limx → 2
(x +1)= −
1
3
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5.2 Limit of a Function5.2 Limit of a Function
L7. The limit of a function raised to an exponent is the limit of the function raised to that exponent.
Example. €
limx → a
f (x)[ ]n
= limx → a
f (x)[ ]n
€
limx → 2
x 2 −1( )3
= limx → 2
x 2 −1( )[ ]3
= 33 = 27
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5.2 Limit of a Function5.2 Limit of a Function
Example. Find .
€
limx → 2
x 2 + 8
x 3 − 4
⎛
⎝ ⎜
⎞
⎠ ⎟
3
€
limx → 2
x 2 + 8
x 3 − 4
⎛
⎝ ⎜
⎞
⎠ ⎟
3
€
= limx → 2
x 2 + 8
x 3 − 4
⎛
⎝ ⎜
⎞
⎠ ⎟
3
€
=limx → 2
x 2 + 8( )
limx → 2
x 3 − 4( )
⎡
⎣
⎢ ⎢
⎤
⎦
⎥ ⎥
3
€
=12
4
⎛
⎝ ⎜
⎞
⎠ ⎟3
€
=27
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5.2 Limit of a Function5.2 Limit of a Function
Example. Find .
€
limx → 2
x 2 − 4
x − 2
€
limx → 2
x 2 − 4
x − 2
€
=limx → 2
(x + 2)(x − 2)
x − 2
€
=limx → 2
(x + 2)
€
=4
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5.2 Limit of a Function5.2 Limit of a Function
Example. Given Find
..
€
limx → 2
f (x)
€
f (x) =x 3 − 4 for x < 2
2x for x ≥ 2
⎧ ⎨ ⎩
€
limx → 2−
f (x)
€
limx → 2+
f (x)
€
= limx → 2−
(x 3 − 4)
€
=4
€
= limx → 2+
2x
€
=4
€
limx → 2
f (x) = 4
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5.2 Limit of a Function5.2 Limit of a Function
1.a Find .
€
limx → 3
(x 2 + 7x − 5)
€
limx → 3
(x 2 + 7x − 5)
€
=32 + 7(3) − 5
€
=25
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5.2 Limit of a Function5.2 Limit of a Function
1.d Find .
€
limx → −2
(x 4 − 3x 3 + 4 x − 5)
€
limx → −2
(x 4 − 3x 3 + 4 x − 5)
€
=(−2)4 − 3(−2)3 + 4(−2) − 5
€
=27
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5.2 Limit of a Function5.2 Limit of a Function
2.c Find .
€
limx → 3 / 2
4x 2 − 9
2x + 3
€
limx → 3 / 2
4x 2 − 9
2x + 3
€
=4 3
2( )2
− 9
2 32( ) + 3
€
=0
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5.2 Limit of a Function5.2 Limit of a Function
3.a Find .
€
limx → 4
x 2 −16
2x − 8
€
limx → 4
x 2 −16
2x − 8
€
=limx → 4
(x + 4)(x − 4)
2(x − 4)
€
=limx → 4
x + 4
2
€
=4
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5.2 Limit of a Function5.2 Limit of a Function
3.c Find .
€
limx → 2
x 2 − 4
x 2 − x − 2
€
limx → 2
x 2 − 4
x 2 − x − 2
€
=limx → 2
(x + 2)(x − 2)
(x − 2)(x +1)
€
=limx → 2
x + 2
x +1
€
=4
3
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5.2 Limit of a Function5.2 Limit of a Function
4.b Find .
€
limx → 2
x 3 + 2x + 3
x 2 + 5
€
limx → 2
x 3 + 2x + 3
x 2 + 5
€
= limx → 2
x 3 + 2x + 3
x 2 + 5
€
=15
9
€
=15
9
€
=13 15
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5.2 Limit of a Function5.2 Limit of a Function
5.a Suppose and .
Find ..€
limx → 4
f (x) = 0
€
limx → 4
g(x) = −2
€
limx → 4
(g(x) + 2)
€
limx → 4
(g(x) + 2)
€
=limx → 4
g(x) + limx → 4
2
€
=−2 + 2
€
=0
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5.2 Limit of a Function5.2 Limit of a Function
7. Given . Find
..
€
limx → 2
f (x)
€
f (x) =x 5 −12 for x < 2
(x + 3)3 − 8 for x ≥ 2
⎧ ⎨ ⎩
€
limx → 2−
f (x)
€
limx → 2+
f (x)
€
= limx → 2−
(x 5 −12)
€
=20
€
= limx → 2+
(x + 3)3 − 8[ ]
€
=19
€
limx → 2
f (x) does not exist.