Download fx( )= L lim fx( )+ gx( )

Math 1016: Elementary Calculus with Trigonometry I
Ch. 3 Introduction to the Derivative
Sec. 3.3: Limits: Algebraic Viewpoints
I.
The Limit Laws
A. Limit Laws
If L, M, c and
and
k are real numbers and let f and g be any 2 functions st lim f ( x ) = L
x→c
lim g ( x ) = M , then
x→c
1. Sum Rule:
lim ⎡⎣ f ( x ) + g ( x ) ⎤⎦ = lim f ( x ) + lim g ( x ) = L + M
x→c
x→c
x→c
2. Difference Rule:
3. Product Rule:
lim ⎡⎣ f ( x ) − g ( x ) ⎤⎦ = lim f ( x ) − lim g ( x ) = L − M
x→a
x→a
x→a
lim ⎡⎣ f ( x ) g ( x )⎤⎦ = lim f ( x ) ⋅ lim g ( x ) = L M
x→c
x→c
x→c
4. Constant Multiple Rule:
5. Quotient Rule:
lim ⎡⎣ k f ( x )⎤⎦ = k lim f ( x ) = k L
x→c
x→c
f ( x) L
⎡ f ( x ) ⎤ lim
x→c
lim ⎢
=
=
, M ≠0
⎥
x→c
g
x
lim
g
x
M
(
)
(
)
⎣
⎦ x→c
r and s are integers with no common factors and s ≠ 0 , then
r
r
r
s
(r)
lim ⎡⎣ f ( x ) ⎤⎦ s = ⎡ lim f ( x ) ⎤ = Ls , provided that Ls is a real number. (If s is even,
⎣ x→c
⎦
x→c
L>0.)
6. If
B. Other Common Limit Rules
1. Constant Function:
2. Identity Function:
lim k = k
x→c
lim x = c
x→c
3. If
n is a positive integer, then lim x n = c n
4. If
n is a positive integer, then lim
5. If
m and b are real numbers, then lim [mx + b] = mc + b .
x→c
x→c
n
x = n c , provided c > 0 if n is even.
x→c
C. Limits of Polynomials
P(x) is a polynomial, P(x) = an x n + an −1 x n −1 + ... + a2 x 2 + a1 x + a0 , then
lim P(x) = P(c) .
If
x→c
Example:
lim(7x 3 − 8) = 7(2)3 − 8 = 7(8) − 8 = 56 − 8 = 48
x→2
D. Limits of Rational Functions
1. c is in the domain of Q(x) : If
P ( x ) P (c)
=
.
x→c Q(x)
Q(c)
P(x) and Q(x) are polynomials and Q(c) ≠ 0 , then
lim
Example:
2.
⎛ x 2 − 4 ⎞ 1− 4 −3
lim ⎜
=
=
= −1
x→1 ⎝ x + 2 ⎟
⎠ 1+ 2 3
c is NOT in the domain of Q(x) :
If
P(x) and Q(x) are polynomials and Q(c) = 0 ,
then use factor, graphical approach or table.
⎛ x2 − 4 ⎞
0
⎛ (x + 2)(x − 2) ⎞
lim ⎜
→ = lim ⎜
lim (x − 2) = −4
⎟⎠ = x→−2
⎟
x→−2 ⎝ x + 2 ⎠
0 x→−2 ⎝
x+2
Example: a.
fHxL=
1
x2
y
b.
k
⎛ 1⎞
lim ⎜ 2 ⎟ → → ∞ ; DNE .
x→0 ⎝ x ⎠
0
500
400
300
200
100
-1.0 -0.5
x 2 − y 2 = (x − y)(x + y)
Helpful Factoring:
x 3 − y 3 = (x − y)(x 2 + xy + y 2 )
x 3 + y 3 = (x − y)(x 2 − xy + y 2 )
E. Examples
Evaluate the following limits:
1.
2.
3.
4.
lim 4 =
x→3
lim x =
x→3
lim (t + 9) =
t→2
lim (5w − 6) =
2
w →2
⎧⎪ x 2 − 5x + 4
; x≠4
5. lim f ( x ) if f (x ) = ⎨
x−4
x →4
⎪⎩ x - 5
; x=4
0.0
0.5
1.0
x
II.
One-Sided Limits
A. Informal Definitions
1. Left Hand Limit / Limit from Below:
lim f ( x ) = L and say the left-handed limit of f ( x ) as x
a. Defn: We write
approaches
x→c −
c [or the limit of f ( x ) as x approaches c from the left] is equal to
L if we can make the values of f ( x ) arbitrarily close to L by taking x to be
sufficiently close to
b. This means that
c and x less than c.
x approaches c from smaller values.
2. Right Hand Limit / Limit from Above:
a. Defn: We write
approaches
to
lim f ( x ) = L and say the right-handed limit f ( x ) as x
x→c +
c [or the limit of f ( x ) as x approaches c from the right] is equal
L if we can make the values of f ( x ) arbitrarily close to L by taking x to be
sufficiently close to
b. This means that
B. Theorem 6
A function
c and x greater than c.
x approaches c from larger values.
f ( x ) has a limit as x approaches c iff it has left-hand and right-hand limits
there and these one-sided limits are equal:
lim f ( x ) = L ⇔ lim− f ( x ) = lim+ f ( x ) = L , where c, L are real numbers.
x→c
x→c
C. Examples
1. Given
x→c
⎧3t + 9 ; t < −2
g(t) = ⎨ 2
⎩t − 1 ; t > −2
a.
lim g(t ) =
b.
lim g(t ) =
t→ 4
t →- 2
lim
x −1
=
x −1
lim
x =
lim
x=
5.
lim
x=
6.
⎛ 1⎞
lim+ ⎜ ⎟ =
x→0 ⎝ x ⎠
7.
⎛ 1⎞
lim− ⎜ ⎟ =
x→0 ⎝ x ⎠
2.
3.
4.
III.
x→1
x→0 +
x→0 −
x→0
Limits as
x → ±∞
In this section we have to modify the definition of a limit so that we can work with infinity.
(Recall that
∞ + ∞ = ∞ but ∞ − ∞ is not a defined operation.)
A. Informal Definitions
1. Defn: Let
f be a fn defined on some interval (a, ∞ ) . Then lim f ( x ) = L means that
x →∞
the values of
2. Defn: Let
f(x) can be made arbitrarily close to L by taking x sufficiently large.
f be a fn defined on some interval (−∞, a) . Then lim f ( x ) = L means
that the values of
large negative.
x →−∞
f(x) can be made arbitrarily close to L by taking x sufficiently
B. Theorem
If
r > 0 is a rational number, then lim
x →∞
defined
∀ x , then lim
x →−∞
D. Examples
Evaluate the following.
1.
2.
lim
x→∞
lim
x→−∞
1
=
x
1
=
x
−4
3.
lim
4.
⎛
1⎞
lim ⎜⎜13 + 3 ⎟⎟ =
y→ −∞ ⎝
y ⎠
5.
⎛ 4 ⎞
⎟⎟ =
lim ⎜⎜
w→ − ∞ ⎝ 8 − 2 ⎠
w
x→∞
( x − 2 )2
=
1
= 0.
xr
1
= 0 . If r > 0 is a rational number s.t. x r is
xr
E. Rational Functions
1. Procedure: When taking the limit at infinity of a rational function in which you get
the form
±∞
, you need to divide all terms by the highest degreed variable in the
±∞
denominator.
2. Examples
a.
⎛ 3x + 7 ⎞
∞
lim ⎜ 2
⎟ → ⇒⇒ ÷ every term by x 2
x →∞⎝ 4x − 2 ⎠
∞
Therefore:
⎛
⎜
⎛ 3x + 7 ⎞
lim ⎜ 2
⎟ = lim
⎜
x→∞ ⎝ 4x − 2 ⎠
x→∞
⎜
⎝
3x 7 ⎞
⎛ 3 7 ⎞ lim ⎛ 3 ⎞ + lim ⎛ 7 ⎞
+ 2 ⎟
+
⎜ ⎟
⎜ ⎟
2
⎜ x x 2 ⎟ x→∞ ⎝ x ⎠ x→∞ ⎝ x 2 ⎠ 0 + 0 0
x
x
= lim
=
=
= =0
2 ⎟
4x 2 2 ⎟⎟ x→∞ ⎜
⎛ 2⎞
4−0 4
4
−
⎜⎝
⎟ lim ( 4 ) − lim ⎜ 2 ⎟
− 2
x→∞ ⎝ x ⎠
x 2 ⎠ x→∞
x2
x ⎠
b. Evaluate the following.
1.)
⎛ 3x + 1 ⎞
lim ⎜
⎝ −7x + 2 ⎟⎠
x→ ∞
⎛ -7x 3 - 8x + 9 ⎞
2.) lim ⎜
x→ -∞ ⎝
2x 2 + 5 ⎟⎠
3. Summary for Rational Functions
If
N 0 = D 0 à The limit lim
If
N 0 < D 0 à The limit lim
If
N 0 > D 0 à The limit lim
x→ ±∞
x→ ±∞
x→ ±∞
( ) = the ratio of the leading coefficients of P & Q.
( ) = 0.
( ) → ±∞
P (x )
Q(x)
P (x )
Q(x)
P (x )
Q(x)
All algebraic work must be shown when calculating limits as
x→±∞!