Chapter 1 Limits and Their Properties (極限與其性質)

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Chapter 1 Limits and Their Properties (極限與其性質)

Hung-Yuan Fan (范洪源)

Department of Mathematics, National Taiwan Normal University, Taiwan

Fall 2018

Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 1, Calculus B 1/122

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本章預定授課範圍

1.2 Functions and Their Graphs 1.3 Inverse Functions

1.4 Exponential and Logarithmic Functions 1.5 Finding Limits Graphically and Numerically 1.6 Evaluating Limits Analytically

1.7 Continuity and One-Sided Limits 1.8 Infinite Limits

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Useful Notations (常用的數學符號)

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1. A set (集合) is a collection of specified objects, and usually denoted by A, B, C, · · · .

R = {x | x is a real number (實數)}

N = {x | x is a positive integer (正整數)}

Z = {x | x is an integer (整數)}

Q = {x | x is a rational number (有理數)}

2. x ∈ A: x belongs to (屬於) A, i.e., x is an element of the set A.

π ∈ R, 5 ∈ N, −7 ∈ Z and 2 3 ∈ Q.

A ⊆ B: A is a subset (子集合) of B, i.e., if x ∈ A, then x ∈ B.

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3. The union (聯集) and intersection (交集) of two sets A and B are defined by

A ∪ B = {x | x ∈ A or x ∈ B}, A ∩ B = {x | x ∈ A and x ∈ B}.

4. ∀: for all (對於所有).

5. ∃: exist (存在).

6. @: does not exist (不存在).

7. s.t. or ∋: such that (使得).

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8. = ⇒: imply that (意指).

9. ⇐⇒: if and only if (若且唯若).

10. Greek letters: (常用希臘字符)

α (alpha), β (beta), γ (gamma), δ (delta), ε (epsilon), θ (theta), λ (lambda), µ (mu), ρ (rho), τ (tau), ϕ (phi), ω (omega), · · ·

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11. The intervals (區間) in R are often denoted by I, e.g., (a, b) = {x ∈ R | a < x < b},

[a, b] = {x ∈ R | a ≤ x ≤ b}, (a, b] = {x ∈ R | a < x ≤ b}, [a, b) = {x ∈ R | a ≤ x < b}.

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12. i.e.: that is (也就是說).

13. e.g.: for example (舉例來說).

14. w.r.t.: with respect to (關於).

15. Def: Definition (定義), Thm: Theorem (定理).

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Section 1.2

Functions and Their Graphs (函數與其圖形)

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Def (實值函數的定義) Let X ⊆ R and Y ⊆ R.

(1) A real-valued function f from X to Y, denoted by f : X → Y, is a correspondence (對應) that assigns to (指派) each x ∈ X one unique (唯一的) value y ∈ Y.

(2) The set X = dom(f) is called the domain (定義域) of f.

(3) The subset of Y defined by

range(f) = {y ∈ Y | ∃ x ∈ X s.t. y = f(x)}

is called the range (值域) of f.

(4) x is the independent variable (自變數) and y is the dependent variable (應變數) of f.

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函數映射的示意圖

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Basic Operations of Functions

Def (函數的基本運算; 1/2)

Let f and g be real-valued functions defined on X ⊆ R.

(1) The sum f + g of f and g is defined by

(f + g)(x) = f(x) + g(x) ∀ x ∈ X.

(2) The difference f − g of f and g is defined by (f − g)(x) = f(x) − g(x) ∀ x ∈ X.

(3) For any k ∈ R, the constant multiple kf of f is defined by (kf)(x) = k · f(x) ∀ x ∈ X.

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Basic Operations of Functions

Def (函數的基本運算; 2/2)

(4) The product fg of f and g is defined by (fg)(x) = f(x) · g(x) ∀ x ∈ X.

(5) The quotient fg of f and g is defined by

( f

g )(x) = f(x)

g(x) ∀ x ∈ X, provided that g(x) ̸= 0 ∀ x ∈ X .

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Three Categories of Elementary Functions (1/2)

1. Algebraic Functions (代數函數)

(a) polynomial (function) of nth degree (n ∈ N)

f(x) = a _n x ⁿ + a n −1 x ⁿ ⁻¹ + · · · + a 0 with a _n ̸= 0.

(b) rational function (有理函數)

f(x) = p(x)/q(x), where p(x) and q(x) ̸= 0 are polynomials.

(c) radical function (根式函數) f(x) = x ^1/n = √ ⁿ

x with n ∈ N.

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Note (根式函數的定義域與值域) Let f(x) = √ ⁿ

x with n ∈ N.

When n is odd (奇數), we know that

dom(f) = range(f) = ( −∞, ∞) = R.

When n is even (偶數), we see that

dom(f) = range(f) = [0, ∞).

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Three Categories of Elementary Functions (2/2)

2. Trigonometric Functions (三角函數)

f(x) = sin x, cos x, tan x, cot x, sec x, csc x.

3. Exponential and Logarithmic Functions (指數與對數函數)

f(x) = a ^x or f(x) = log _a x, where 0 < a ̸= 1 . (See Section 1.4 later!)

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Def (合成函數的定義)

Let X, Y and Z be subsets of R. The composite function (合成函數) of f : Y → Z and g : X → Y is defined by

(f ◦ g)(x) = f(g(x)) ∀ x ∈ X.

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Def (Even and Odd Functions)

Let f be a real-valued function defined on X ⊆ R.

(1) f is an even function (偶函數) if f( −x) = f(x) ∀ x ∈ X . (2) f is an odd function (奇函數) if f( −x) = −f(x) ∀ x ∈ X .

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Notes

The graph of an even function is symmetric w.r.t. the y-axis.

(偶函數的圖形對稱於 y-軸)

The graph of an odd function is symmetric w.r.t. the origin.

(奇函數的圖形對稱於原點)

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Section 1.3 Inverse Functions

(反函數)

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Def (反函數的定義)

Let f : X → Y be a function, where X, Y ⊆ R. A function g : Y → X is called the inverse function (反函數) of f if

g(f(x)) = x ∀ x ∈ X and f(g(y)) = y ∀ y ∈ Y.

In this case, we denote g = f ⁻¹ . (讀作 f-inverse)

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Remarks

1

If f ⁻¹ ∃, then (f ⁻¹ ) ⁻¹ = f.

2

In general, it is true that f ⁻¹ (x) ̸= _f(x) ¹ .

3

The graph of f ⁻¹ is a reflection (反射) of the graph of f in the line y = x, i.e., b = f(a) ⇐⇒ f ⁻¹ (b) = a.

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Main Questions

Does f always have an inverse function f ⁻¹ ? When does f ⁻¹ exist for any real-valued function f?

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Def (一對一函數的定義)

Let f be a real-valued function defined on X ⊆ R. If f(x ₁ ) ̸= f(x 2 ) for any x ₁ ̸= x 2 ∈ X , then f is an one-to-one function. (一對一函數; 簡寫成 1-1)

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Existence of f ⁻¹

Thm (反函數 f ⁻¹ 的存在性)

Let f be a real-valued function defined on X ⊆ R. Then f ⁻¹ ∃ ⇐⇒ f is one-to-one on X.

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示意圖 (承上頁)

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Inverse Trigonometric Functions (反三角函數)

In order to obtain the inverse trigonometric functions, we need to restrict the domains of six trigonometric functions.

Conventionally, the following functions sin : [ −π

2 , π

2 ] → [−1, 1], cos : [0, π] → [−1, 1]

tan : ( −π 2 , π

2 ) → (−∞, ∞), cot : (0, π) → (−∞, ∞) sec : [0, π

2 ) ∪ ( π

2 , π] → (−∞, −1] ∪ [1, ∞), csc : [ −π

2 , 0) ∪ (0, π

2 ] → (−∞, −1] ∪ [1, ∞) are both 1-1 on the restricted domains.

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Section 1.4

Exponential and Logarithmic Functions (指數函數與對數函數)

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Def (以 a 為底的指數函數)

The exponential function with base number a is defined by f(x) = a ^x ∀ x ∈ R,

where 0 < a ̸= 1 .

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Thm (Basic Properties of a ^x ) Let f(x) = a ^x with 0 < a ̸= 1. Then

1

dom(f) = R = (−∞, ∞).

2

range(f) = (0, ∞), i.e., f(x) = a ^x > 0 ∀ x ∈ R .

3

f(0) = a ⁰ = 1, i.e., the y-intercept of f is (0, 1), and f(1) = a.

4

f is one-to-one on R, i.e., f ⁻¹ : (0, ∞) → R ∃ .

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Thm (Laws of Exponents; 指數律) If a, b > 0 and x, y ∈ R, then

1

a ^x a ^y = a ^x+y .

2

(a ^x ) ^y = a ^xy = (a ^y ) ^x .

3

(ab) ^x = a ^x b ^x .

4

a ^x

a ^y = a ^x ^−y .

5

( a b

) x

= a b ^x ^x .

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Def (Euler’s number; 歐拉數或尤拉數) The irrational number e = lim

x →0 (1 + x) ^1/x ≈ 2.71828182846 · · · . Note: see Example 3 for observing the behavior of

f(x) = (1 + x) ^1/x as x approaches 0.

Definition of e ^x

For the base number a = e > 1, f(x) = a ^x = e ^x is called the natural exponential function (自然指數函數).

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The Inverse Function of e ^x

Since the function f(x) = e ^x is one-to-one on R, it must have an inverse function f ⁻¹ : (0, ∞) −→ R = (−∞, ∞)!

Definition of ln x

The inverse function of f(x) = e ^x , denoted by f ⁻¹ (x) = ln x, is called the natural logarithmic function (自然對數函數). Moreover, we have

y = ln x ∀ x > 0 ⇐⇒ e ^y = x ∀ y ∈ R.

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Thm (Basic Properties of ln x) Let f(x) = ln x. Then

1

dom(f) = (0, ∞).

2

range(f) = R = (−∞, ∞).

3

f(1) = ln1 = 0, i.e., the x-intercept of f is (1, 0), and f(e) = ln e = 1.

4

f is one-to-one on (0, ∞), i.e., f ⁻¹ : R → (0, ∞) ∃ .

5

ln(e ^x ) = x for x ∈ R , and e ^{ln x} = x for x > 0.

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示意圖 (承上頁)

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Thm (Laws of Logarithms; 對數律) If x > 0, y > 0 and z ∈ R, then

1

ln(xy) = ln x + ln y.

2

ln ( x y

)

= ln x − ln y.

3

ln(x ^z ) = z · ln x.

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Section 1.5

Finding Limits Graphically and Numerically

(從圖形上與數值上求極限)

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Informal Definition of a Limit

Def (函數極限值的非正式定義)

Let f be a real-valued function defined on X ⊆ R with c ∈\X . If f(x) becomes arbitrarily close to a unique number L ∈ R as x

approaches c from either side, then the limit of f is L as x approaches c, denoted by lim

x →c f(x) = L.

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Nonexistence of a Limit

Type I (在 c 點兩邊的極限值不相等)

If f(x) → L 1 and f(x) → L 2 , where L ₁ ̸= L 2 , as x approaches c from either side, then lim

x →c f(x) @ .

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示意圖 (承上例)

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Nonexistence of a Limit

Type II (函數值在 c 點附近無上下界)

If f(x) increases (遞增) or decreases (遞減) without bound as x approaches c from either side, then lim

x →c f(x) @ .

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Nonexistence of a Limit

Type III (函數值在 c 點附近震盪)

If f(x) oscillates (震盪) between two fixed values as x approaches c from either side, then lim

x →c f(x) @ .

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示意圖 (承上例)

Note: 函數值在 −1 和 1 之間來回 [振盪，但沒碰到 y-軸喔!

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Formal Definition of a Limit

Def (函數極限值的正式定義)

Let f be a real-valued function defined on X ⊆ R with c ∈\X . Then

x lim →c f(x) = L.

⇐⇒∀ε > 0, ∃ δ > 0 s.t. if 0 < |x − c| < δ (and x ∈ X), then |f(x) − L| < ε.

Note: this is also called the ε-δ definition of a limit.

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極限的正式定義

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Section 1.6

Evaluating Limits Analytically (從分析上求極限)

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Thm (Basic Limit Laws; 1/2)

Let b, c ∈ R and let f and g be real-valued functions with

x lim →c f(x) = L, lim

x →c g(x) = K.

(1) lim

x →c b = b and lim

x →c |x| = |c|.

(2) lim

x →c x ⁿ = c ⁿ and lim

x →c

[ f(x) ] n

= L ⁿ ∀ n ∈ N.

(3) lim

x →c [b · f(x)] = b · [

lim x →c f(x) ]

= b · L.

(4) lim

x →c

[

f(x) ± g(x) ]

= [

lim x →c f(x) ]

± [

x lim →c g(x) ]

= L ± K.

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Thm (Basic Limit Laws; 2/2) (5) lim

x →c

[ f(x) · g(x) ]

= [ lim

x →c f(x) ]

· [

lim x →c g(x) ]

= L · K.

(6) lim

x →c

f(x) g(x) =

x lim →c f(x)

x lim →c g(x) = L K if K ̸= 0 . (7) The Limit of f ◦ g: (合成函數的極限值)

If lim

x →c f(x) = f(K), then lim

x →c f(g(x)) = f (

lim x →c g(x) )

= f(K).

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Thm (Limits of Elementary Functions; 1/2)

Let c be a real number in the domain of the given function.

(1) If p(x) is a polynomial, then lim

x →c p(x) = p(c).

(2) If r(x) = p(x)/q(x) is a rational function with q(c) ̸= 0 , then

x lim →c r(x) = r(c) = p(c)/q(c).

(3) lim

x →c

√ n

x = √ ⁿ

c ∀ n ∈ N, where c ≥ 0 when n is even and c ∈ R when n is odd.

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Thm (Limits of Elementary Functions; 2/2) (4) Limits of 6 trigonometric functions are given by

x lim →c sin x = sin c, lim

x →c cos x = cos c, lim

x →c tan x = tan c,

x lim →c cot x = cot c, lim

x →c sec x = sec c, lim

x →c csc x = csc c.

(5) lim

x →c a ^x = a ^c for a > 0 and c ∈ R.

(6) lim

x →c ln x = ln c for c > 0.

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Thm 1.7 (化簡函數後求極限值)

If ∃ δ > 0 s.t. f(x) = g(x) ∀ x ∈ (c − δ, c) ∪ (c.c + δ) , then

x lim →c f(x) = lim

x →c g(x).

Note: 將 f(x) 簡化為 g(x) 後，兩者在 c 點附近的極限值相等!

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Thm 1.8 (Squeeze (or Sandwich) Thm; 夾擠定理或夾擊定理) If ∃ δ > 0 s.t. h(x) ≤ f(x) ≤ g(x) ∀ x ∈ (c − δ, c) ∪ (c, c + δ) , and

x lim →c h(x) = L = lim

x →c g(x), then lim

x →c f(x) = L.

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Thm 1.8 的示意圖

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Thm 1.9 (Some Special Limits)

1

lim

θ →0

sin θ θ = 1.

2

lim

θ →0

1 − cos θ θ = 0.

3

lim

x →0 (1 + x) ^1/x = e.

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Section 1.7

Continuity and One-Sided Limits (連續性與單邊極限)

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Continuity of a Function

Def (實值函數的連續性)

Let f be a real-valued function defined on I = (a, b) with c ∈ I . (1) f is continuous (連續的; 簡寫為 conti.) at c if lim

x →c f(x) = f(c).

(2) f is conti. on I if it is conti. at each c ∈ I.

(3) f is everywhere conti. (處處連續) if it is conti. on R = (−∞, ∞).

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Discontinuity of a Function

Def (函數的不連續性)

Let f be a real-valued function defined on I = (a, b) with c ∈ I.

(1) f has a discontinuity (不連續點; 簡寫為 disconti.) at c if it is NOT conti. at c.

(2) A disconti. of f at c is called removable (可移除的) if f can be made conti. at c by redefining f(c). Otherwise, the disconti. at c is called nonremovable (不可移除的).

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示意圖 (承上例)

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One-Sided Limits

Def (單邊極限值的定義)

(1) f has the limit L from the right (or the right-hand limit L; 右極限值) at c, denoted by lim

x →c ⁺ f(x) = L, if f(x) → L as x → c from the right.

(2) f has the limit L from the left (or the left-hand limit L; 左極限值) at c, denoted by lim

x →c ⁻ f(x) = L, if f(x) → L as x → c from the left.

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單邊極限值的示意圖

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Solution of Example 3

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Existence of a Limit

Thm 1.10 (函數極限值存在的等價條件)

x lim →c f(x) = L ⇐⇒ lim

x →c ⁻ f(x) = L = lim

x →c ⁺ f(x).

(f 在 c 點的極限值為 L ⇐⇒ 左右極限值均為 L)

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One-Sided Continuity

Def (單邊連續的定義)

Let f be a real-valued function defined on X ⊆ R with c ∈ X.

(1) f is conti. from the right (佑蓮續) at c if lim

x →c ⁺ f(x) = f(c).

(2) f is conti. from the left (左蓮續) at c if lim

x →c ⁻ f(x) = f(c).

Remark

f is conti. at c ⇐⇒ f is conti. from the right and from the left at c.

(f 在 c 點連續 ⇐⇒ f 在 c 點右蓮續且左連續)

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Continuity on a Closed Interval

Def (在閉區間上的連續性)

We say that f is conti. on I = [a, b] if the following conditions hold:

1

f is conti. on the open interval (a, b).

2

f is conti. from the right at a, i.e., lim

x →a ⁺ f(x) = f(a).

3

f is conti. from the left at b, i.e., lim

x →b ⁻ f(x) = f(b).

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Properties of Continuity

Thm (連續函數的性質)

1

If f and g are conti. at c and b ∈ R, then f ± g, bf, fg and f/g with g(c) ̸= 0 are conti. at c, respectively.

2

If g is conti. at c and f is conti. at g(c), then (f ◦ g)(x) = f(g(x)) is conti. at c.

3

All elementary functions are conti. on their domains.

Note: the above properties are also true for one-sided continuity!

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示意圖 (承上例)

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Intermediate Value Theorem

Thm 1.13 (I.V.T.; 中間值定理)

If f is conti. on [a, b], f(a) ̸= f(b) and k is any number between f(a) and f(b), then ∃ c ∈ [a, b] s.t. f(c) = k.

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Section 1.8 Infinite Limits

(無窮極限)

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Def (無窮極限值的定義; 1/2) (1) lim

x →c f(x) = ∞ ⇐⇒ ∀ M > 0, ∃ δ > 0 s.t. if 0 < |x − c| < δ, then f(x) > M.

(2) lim

x →c f(x) = −∞ ⇐⇒ ∀ N < 0, ∃ δ > 0 s.t. if 0 < |x − c| < δ, then f(x) < N.

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示意圖 (承上頁)

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Def (無窮極限值的定義; 2/2) (3) lim

x →c ⁺ f(x) = ∞ ( or lim

x →c ⁻ f(x) = ∞ )

⇐⇒ ∀ M > 0, ∃ δ > 0 s.t. if c < x < c + δ (or c − δ < x < c ), then f(x) > M.

(4) lim

x →c ⁺ f(x) = −∞ ( or lim

x →c ⁻ f(x) = −∞ )

⇐⇒ ∀ N < 0, ∃ δ > 0 s.t. if c < x < c + δ (or c − δ < x < c ), then f(x) < N.

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示意圖 (承上頁)

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重要口訣 (切記!)

若 +0 與 −0 分別代表接近零的正數與負數, 則

1 +0 = ∞, ₋₀ ¹ = −∞

∞ 1 = _−∞ ¹ = 0,

其中，+ ∞ = ∞ 和 −∞ 分別為正負無窮遠處的符號。

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Vertical Asymptotes

Def (鉛直漸近線或垂直漸近線) If lim

x →c ⁺ f(x) = ±∞ or lim

x →c ⁻ f(x) = ±∞, then the line x = c is a vertical asymptote (垂直漸近線) of the graph of f. 　

Thm 1.14 (判斷垂直漸近線的位置)

If f and g are conti. on an open interval I containing c, where g(x) ̸= 0 ∀ x ∈ I\{c}. If f(c) ̸= 0 and g(c) = 0, then f(x)

g(x) has a vertical asymptote at x = c.

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Example 3 的示意圖 (承上頁)

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Thm 1.15 (Properties of Infinite Limits) Suppose that lim

x →c f(x) = ±∞ and lim _x

→c g(x) = L ̸= 0.

1

lim

x →c [f(x) ± g(x)] = ±∞.

2

lim

x →c [f(x)g(x)] = ±∞ if L > 0.

3

lim

x →c [f(x)g(x)] = ∓∞ if L < 0.

4

lim

x →c

g(x) f(x) = 0.

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Chapter 1 Limits and Their Properties (極限與其性質)

Chapter 1

Limits and Their Properties (極限與其性質)

Hung-Yuan Fan (范洪源)

Department of Mathematics, National Taiwan Normal University, Taiwan

Fall 2018

本章預定授課範圍

1.2 Functions and Their Graphs 1.3 Inverse Functions

1.4 Exponential and Logarithmic Functions 1.5 Finding Limits Graphically and Numerically 1.6 Evaluating Limits Analytically

1.7 Continuity and One-Sided Limits 1.8 Infinite Limits

Useful Notations (常用的數學符號)

1. A set (集合) is a collection of specified objects, and usually denoted by A, B, C, · · · .

R = {x | x is a real number (實數)}

N = {x | x is a positive integer (正整數)}

Z = {x | x is an integer (整數)}

Q = {x | x is a rational number (有理數)}

2. x ∈ A: x belongs to (屬於) A, i.e., x is an element of the set A.

π ∈ R, 5 ∈ N, −7 ∈ Z and 2 3 ∈ Q.

A ⊆ B: A is a subset (子集合) of B, i.e., if x ∈ A, then x ∈ B.

3. The union (聯集) and intersection (交集) of two sets A and B are defined by

A ∪ B = {x | x ∈ A or x ∈ B}, A ∩ B = {x | x ∈ A and x ∈ B}.

4. ∀: for all (對於所有).

5. ∃: exist (存在).

6. @: does not exist (不存在).

7. s.t. or ∋: such that (使得).

8. = ⇒: imply that (意指).

9. ⇐⇒: if and only if (若且唯若).

10. Greek letters: (常用希臘字符)

α (alpha), β (beta), γ (gamma), δ (delta), ε (epsilon), θ (theta), λ (lambda), µ (mu), ρ (rho), τ (tau), ϕ (phi), ω (omega), · · ·

11. The intervals (區間) in R are often denoted by I, e.g., (a, b) = {x ∈ R | a < x < b},

[a, b] = {x ∈ R | a ≤ x ≤ b}, (a, b] = {x ∈ R | a < x ≤ b}, [a, b) = {x ∈ R | a ≤ x < b}.

12. i.e.: that is (也就是說).

13. e.g.: for example (舉例來說).

14. w.r.t.: with respect to (關於).

15. Def: Definition (定義), Thm: Theorem (定理).

Section 1.2

Functions and Their Graphs (函數與其圖形)

Def (實值函數的定義) Let X ⊆ R and Y ⊆ R.

(1) A real-valued function f from X to Y, denoted by f : X → Y, is a correspondence (對應) that assigns to (指派) each x ∈ X one unique (唯一的) value y ∈ Y.

(2) The set X = dom(f) is called the domain (定義域) of f.

(3) The subset of Y defined by

range(f) = {y ∈ Y | ∃ x ∈ X s.t. y = f(x)}

is called the range (值域) of f.

(4) x is the independent variable (自變數) and y is the dependent variable (應變數) of f.

函數映射的示意圖

Basic Operations of Functions

Def (函數的基本運算; 1/2)

Let f and g be real-valued functions defined on X ⊆ R.

(1) The sum f + g of f and g is defined by

(f + g)(x) = f(x) + g(x) ∀ x ∈ X.

(2) The difference f − g of f and g is defined by (f − g)(x) = f(x) − g(x) ∀ x ∈ X.

(3) For any k ∈ R, the constant multiple kf of f is defined by (kf)(x) = k · f(x) ∀ x ∈ X.

Basic Operations of Functions

Def (函數的基本運算; 2/2)

(4) The product fg of f and g is defined by (fg)(x) = f(x) · g(x) ∀ x ∈ X.

(5) The quotient fg of f and g is defined by

( f

g )(x) = f(x)

g(x) ∀ x ∈ X, provided that g(x) ̸= 0 ∀ x ∈ X .

Three Categories of Elementary Functions (1/2)

1. Algebraic Functions (代數函數)

(a) polynomial (function) of nth degree (n ∈ N)

f(x) = a n x n + a n −1 x n −1 + · · · + a 0 with a n ̸= 0.

(b) rational function (有理函數)

f(x) = p(x)/q(x), where p(x) and q(x) ̸= 0 are polynomials.

(c) radical function (根式函數) f(x) = x 1/n = √ n

x with n ∈ N.

Note (根式函數的定義域與值域) Let f(x) = √ n

x with n ∈ N.

When n is odd (奇數), we know that

dom(f) = range(f) = ( −∞, ∞) = R.

When n is even (偶數), we see that

dom(f) = range(f) = [0, ∞).

Three Categories of Elementary Functions (2/2)

2. Trigonometric Functions (三角函數)

f(x) = sin x, cos x, tan x, cot x, sec x, csc x.

3. Exponential and Logarithmic Functions (指數與對數函數)

f(x) = a x or f(x) = log a x, where 0 < a ̸= 1 . (See Section 1.4 later!)

Def (合成函數的定義)

Let X, Y and Z be subsets of R. The composite function (合成函 數) of f : Y → Z and g : X → Y is defined by

f(x) = a _n x ⁿ + a n −1 x ⁿ ⁻¹ + · · · + a 0 with a _n ̸= 0.

(c) radical function (根式函數) f(x) = x ^1/n = √ ⁿ

Note (根式函數的定義域與值域) Let f(x) = √ ⁿ

f(x) = a ^x or f(x) = log _a x, where 0 < a ̸= 1 . (See Section 1.4 later!)

Let X, Y and Z be subsets of R. The composite function (合成函數) of f : Y → Z and g : X → Y is defined by

In this case, we denote g = f ⁻¹ . (讀作 f-inverse)

If f ⁻¹ ∃, then (f ⁻¹ ) ⁻¹ = f.

In general, it is true that f ⁻¹ (x) ̸= _f(x) ¹ .

The graph of f ⁻¹ is a reflection (反射) of the graph of f in the line y = x, i.e., b = f(a) ⇐⇒ f ⁻¹ (b) = a.

Does f always have an inverse function f ⁻¹ ? When does f ⁻¹ exist for any real-valued function f?

Let f be a real-valued function defined on X ⊆ R. If f(x ₁ ) ̸= f(x 2 ) for any x ₁ ̸= x 2 ∈ X , then f is an one-to-one function. (一對一函數; 簡寫成 1-1)

Existence of f ⁻¹

Thm (反函數 f ⁻¹ 的存在性)

Let f be a real-valued function defined on X ⊆ R. Then f ⁻¹ ∃ ⇐⇒ f is one-to-one on X.

The exponential function with base number a is defined by f(x) = a ^x ∀ x ∈ R,

Thm (Basic Properties of a ^x ) Let f(x) = a ^x with 0 < a ̸= 1. Then

range(f) = (0, ∞), i.e., f(x) = a ^x > 0 ∀ x ∈ R .

f(0) = a ⁰ = 1, i.e., the y-intercept of f is (0, 1), and f(1) = a.

f is one-to-one on R, i.e., f ⁻¹ : (0, ∞) → R ∃ .

a ^x a ^y = a ^x+y .

(a ^x ) ^y = a ^xy = (a ^y ) ^x .

(ab) ^x = a ^x b ^x .

a ^x

a ^y = a ^x ^−y .

= a b ^x ^x .

x →0 (1 + x) ^1/x ≈ 2.71828182846 · · · . Note: see Example 3 for observing the behavior of

f(x) = (1 + x) ^1/x as x approaches 0.

Definition of e ^x

For the base number a = e > 1, f(x) = a ^x = e ^x is called the natural exponential function (自然指數函數).

The Inverse Function of e ^x

Since the function f(x) = e ^x is one-to-one on R, it must have an inverse function f ⁻¹ : (0, ∞) −→ R = (−∞, ∞)!

The inverse function of f(x) = e ^x , denoted by f ⁻¹ (x) = ln x, is called the natural logarithmic function (自然對數函數). Moreover, we have

y = ln x ∀ x > 0 ⇐⇒ e ^y = x ∀ y ∈ R.