○○○ 5.1.1 Definition⎾f(x)⏐x￫∞⏌= L ⪢⪢
Let f:A￫R be a function whose domain A contains the interval [a,∞) for some fixed real value of a. Then f(x) tends to L as x tends to infinity, if for every ε>0 there exists a real number X such that x>X ⇒ ⎹f(x)−L⎸<ε.
39169#5002 SBLNGS CHLDRN 5002

○○○ 5.1.2 Definition⎾f(x)︱x￫a⏌= L ⪢⪢
f(x) tends to L as x tends to a, if and only if for every ε>0 there exists a δ>0 such that 0<⎹x−a⎸<δ ⇒ ⎹f(x)−L⎸<ε.
39169#5003 SBLNGS CHLDRN 5003

○○○ 5.1.3 Rules ⪢⪢
Suppose that ⎾f(x)⏐x￫a⏌= L and ⎾g(x)⏐x￫a⏌= M. Then the following rules apply. ❶ Sum Rule ⎾f(x)+g(x)⏐x￫a⏌= L+M ❷ Product Rule ⎾f(x)·g(x)⏐x￫a⏌= L·M ❸ Quotient Rule ⎾f(x)/g(x)⏐x￫a⏌= L/M, provided that M≠0.
39169#3803 SBLNGS CHLDRN 3803

○○○ 5.1.4 Sandwich rule ⪢⪢
Let f, g and h be three functions satisfying f(x)≤g(x)≤h(x) in some detected neighborhood of x=a. If ⎾f(x)⏐x￫a⏌=⎾h(x)⏐x￫a⏌= L then ⎾g(x)⏐x￫a⏌= L.
39169#3422 SBLNGS CHLDRN 3422

○○○ 5.1.5 Composite rule ⪢⪢
If ⎾f(x)⏐x￫a⏌= L and g is a function that is continuous in some neighborhood of x=L, then ⎾(g∘f)(x)⏐x￫a⏌= g(L).
39169#3424 SBLNGS CHLDRN 3424

○○○ 5.1.6 Theorm ⪢⪢
⎾f(x)⏐x￫a⏌= L ⇔ ⎾f(x)⏐x￫a⁻⏌=⎾f(x)⏐x￫a⁺⏌= L
39169#3421 SBLNGS CHLDRN 3421

Songs in Easy English

○○○ 5.2.1 Definition ⪢⪢
f is continuous at c if and only if ⎾f(x)⏐x￫c⏌= f(c).
39169#5081 SBLNGS CHLDRN 5081

○○○ 5.2.2 Definition ⪢⪢
f is continuous at c if and only if for every ε>0 there exists δ>0 such that ⎹x−c⎸<δ ⇒ ⎹f(x)−f(c)⎸<ε.
39169#5000 SBLNGS CHLDRN 5000

○○○ 5.2.3 Theorem ⪢⪢
f is continuous at c if and only if f is both left-continuous and right-continuous at c.┆If ⎾f(x)⏐x￫c⁻⏌ exists and equals f(c) then f is left-continuous at c. If ⎾f(x)⏐x￫c⁺⏌ exists and equals f(c) then f is right-continuous at c.
39169#5123 SBLNGS CHLDRN 5123

○○○ 5.2.4 Rules ⪢⪢
Sum rule. If f and g are continuous at c then f+g is continuous at c.┆Product rule. If f and g are continuous at c then f·g is continuous at c.┆Reciprocal rule. If f is continuous at c and f(c) is not 0 then 1/f is continuous at c.
39169#5158 SBLNGS CHLDRN 5158

○○○ 5.2.5 Sandwich rule ⪢⪢
Let f, g and h be functions such that f(x)≤g(x)≤h(x) for all x in some neighborhood of c and such that f(c)=g(c)=h(c). If f and h are conyinuous at c, then so is g.
39169#5160 SBLNGS CHLDRN 5160

○○○ 5.2.6 Composite rule ⪢⪢
Let f and g be continuous at c and f(c) respectively, such that g∘f is defined. Then g∘f is continuous at c.
39169#3734 SBLNGS CHLDRN 3734

○○○ 5.2.7 Inverse rule ⪢⪢
Suppose that f:A￫B is a bisection where A and B are intervals. If f is continuous on A, then f˂ is continuous on B. ┆ A function f:A￫B where A and B are intervals is a bijection if, for every b ∈ B, there exists a unique a ∈ A with f(a) = b.
39169#5163 SBLNGS CHLDRN 5163

○○○ differentiable at c ⪢⪢
f is differentiable at c if and only if ⎾(f(x)−f(c))/(x−c)︱x￫c⏌ exists. The value of the limit, called the derivative of f at c, is denoted by f(x)ᐁx[c].
39169#5001 SBLNGS CHLDRN 5001

• differentiable ⇒ continuous
If f is differentiable at c then f is continuous at c.
39169#9758 SBLNGS CHLDRN 9758