Calculus The Derivative Chapter 3 Section 1 Limits Essential Question: What are the differences in the expressions: and ? Student Objectives: The student will determine the limit of a function using a table of values. The student will determine the limit of a function by factoring. The student will determine the limit of a function by using the conjugate of the numerator or denominator. The student will determine the limit of a function at x approaches ∞ or - ∞. The student will determine the limit of the function from the right or left of a value using “fuzzy math.” Terms: Limit Limit from the left Limit from the right Limits of infinity One-sided limit Piecewise function Two-sided limit lim x→a f x () fa ()
Transcript
CalculusThe Derivative
Chapter 3 Section 1Limits
Essential Question: What are the differences in the expressions: � and � ?Student Objectives: The student will determine the limit of a function using a table of values.
The student will determine the limit of a function by factoring.The student will determine the limit of a function by using the conjugate
of the numerator or denominator.The student will determine the limit of a function at x approaches ∞ or
- ∞.The student will determine the limit of the function from the right or left
of a value using “fuzzy math.”
Terms:Limit
Limit from the left
Limit from the right
Limits of infinity
One-sided limit
Piecewise function
Two-sided limit
limx→a
f x( ) f a( )
Key Concepts:
Rules for LimitsLet a, A, and B be real numbers, and let f and g be functions such that lim
x→a f x( ) = A and lim
x→a g x( ) = B
1. If k is a constant, then lim x→a
k = k and
lim x→a
k ⋅ f x( )⎡⎣ ⎤⎦ = lim x→a
k ⋅ f x( )⎡⎣ ⎤⎦ ⋅ lim x→a k ⋅ f x( )⎡⎣ ⎤⎦ = k ⋅A = kA
2. The limit of a sum or difference is the sum or difference of their limits.
lim x→a
f x( ) ± g x( )⎡⎣ ⎤⎦ = lim x→a
f x( ) ± lim x→a
g x( ) = A ± B
3. The limit of a product is the product of their limits.
lim x→a
f x( ) ⋅g x( )( ) = lim x→a
f x( ) ⋅ lim x→a
g x( ) = A ⋅B = AB
Limit of a FunctionLet f be a function and let a and L be real numbers. If 1. as x takes values closer and closer (but not equal) to a on both sides of a, the corresponding values of f x( ) get closer (and perhaps equal) to L; and 2. the value of f x( ) can be made as close to L as desiredby taking values of x close enough to a;then L is the limit of f x( ) as x approaches a, written lim
x→af x( ) = L
Existence of LimitsThe limit of f as x approaches a may not exist. 1. If f x( ) becomes infinitely large in magnitude (positive or negative) as x approaches the number a from either side, we write lim
x→af x( ) = ∞ or
limx→a
f x( ) = −∞. In either case the limit does not exist.
2. If f x( ) becomes infinitely large in magnitude (positive) as x approaches a from one side and infinitely large in magnitude (negative) as x approaches a from the other side, then lim
x→af x( ) does not exist.
limx→a
f x( ) = −∞. In either case the limit does not exist.
3. If limx→a−
f x( ) = L and limx→a+
f x( ) = M , and L ≠ M, then limx→a
f x( ) does not exist.
Graphing Calculator Skills:
Create a table of values for the given function on either side of a given value.
Rules for Limits......continuedLet a, A, and B be real numbers, and let f and g be functions such that lim
x→a f x( ) = A and lim
x→a g x( ) = B
4. The limit of a quotient is the quotient of their limits, given the limit of the denominator is not equal to zero.
lim x→a
f x( )g x( )
⎛⎝⎜
⎞⎠⎟=
lim x→a
f x( ) lim x→a
g x( ) =AB
, if B ≠ 0
5. If p x( )is a polynomial, then lim x→a
p x( ) = p a( ).
6. For any real number k, lim x→a
f x( )( )k = lim x→a
f x( )( )k = Ak .
7. If lim x→a
f x( ) = lim x→a
g x( )if f x( ) = g x( ), fall x ≠ a.
8. For any real number b such that b > 0, lim x→a
b( ) f x( ) = b( ) lim x→a
f x( ) = b lim x→a
f x( ).
9. For any real number b such that 0 < b < 1,
lim x→a
logb f x( )( ) = logb lim x→a
f x( )( ) = logb A if A > 0.
Limits at Infinity For any positive real number n,
limx→∞
1xn
= 0 and limx→−∞
1xn
= 0.
Finding Limits at Infinity
If f x( ) = p x( )q x( ) , for polynomials p x( ), q x( ), and q x( ) ≠ 0,
limx→∞
f x( ) and limx→−∞
f x( ) can be found as follows.
1. Divide p x( )and q x( ) by the highest power in q x( ). 2. Use the rules of limits, including the rules for limits at infinity to find the limit of the result found from step 1.
Sample Questions:
1. Determine the following limit: �
2. Use a table of values to determine the following limit: �
limx→3
2x2 − 5x +1( )
limx→1
2
2 − 5 − 2x
x − 12
⎛
⎝
⎜⎜⎜
⎞
⎠
⎟⎟⎟
x f(x)
0.49
0.499
0.4999
0.49999
0.5
0.50001
0.5001
0.501
0.51
3. Determine the following limit: �
4. Determine the following limit: �
limx→2
x − 2x2 − 4
⎛⎝⎜
⎞⎠⎟
limx→−3.5
6x2 + 25x +148x2 +18x − 35
⎛⎝⎜
⎞⎠⎟
5. Determine the following limit: � limx→−3
16 − 3x − 52x + 6
⎛
⎝⎜⎞
⎠⎟
6. Use the graph below to determine the following limit:
�
7. Use the graph below to determine the following limit:
�
limx→2−
f (x) = ______, limx→2
f (x) = ______, limx→2+
f (x) = ______
limx→−1−
f (x) = ______, limx→−1
f (x) = ______, limx→−1+
f (x) = ______
8. Determine the following limit: �
9. Determine the following limit: �
limx→∞
5x3 − 8x2 + 6x +12x3 − 7x + 3
⎛⎝⎜
⎞⎠⎟
limx→∞
8x2 + 6x +12x4 − 7x2 + 3
⎛⎝⎜
⎞⎠⎟
10. Determine the following limit: �
11. Determine the following limit: � .
limx→−∞
2x6 − 7x4 − 9x2 + 48x3 − 6x + 3
⎛⎝⎜
⎞⎠⎟
limx→4
x + 35
!
"##
$
%&&
12. Determine the following limit: � .
13. Determine the following limit: � .
limx→−2
11− 5x3
!
"##
$
%&&
limx→5
g x( ), if g x( ) = 3x − 7, x > 52x − 3 x ≤ 5
⎧⎨⎪
⎩⎪
14. Determine the following limit: � .
15. Use “fuzzy math” to determine the following limit: � .
