Evan B.

Direct Substitution Property

If f is a polynomial or a rational function and a is in the domain of f, then:

the limit of f(x) as x approaches a = f(a)

The Squeeze Theorem

If f(x)<=g(x)<=h(x) when x is near a (except possibly at a) and:

the limit of f(x) as x approaches a = the limit of h(x) as x approaches a = L

then:

the limit of g(x) as x approaches a = L

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The Precise Definition of a Limit

Let f be a function defined on some open interval that contains the number a, except possibly at a itself. Then we say that the limit of f(x) as x approaches a is L. and we write:

the limit of f(x) as x approaches a = L

if for every number ε>0 there is a number δ>0 such that:

if 0<|x-a|<δ then |f(x)-L|<ε

Definition of Left-Hand Limit

The limit of f(x) as x approaches x^{-} = L

if for every number ε>0 there is a number δ>0 such that:

if a<x<a+δ then |f(x)-L|<ε

Definition of Right-Hand Limit

The limit of f(x) s x approaches a^{+}=L

if for every number ε>0 there is a number δ>0 such that:

if a<x<a+δ then |f(x)-L|<ε

Definition of Infinite Limits

Let f be a function defined on some open interval that contains number a, except possibly at a itself. Then:

the limit of f(X) as x approaches a = ∞

means that for every positive number M there is a positive number δ such that:

if 0<|x-a|<δ then f(x)>M

Definition of Negative Infinite Limits

Let f be a function defined on some open interval that contains number a, except possibly at a itself. Then:

the limit of f(X) as x approaches a = -∞

means that for every negative number N there is a positive number δ such that:

if 0<|x-a|<δ then f(x)<N

The Intermediate Value Theorem

Suppose that f is continuous on the closed interval [a, b] and let N be any number between f(a) and f(b), where f(a) does not equal f(b). Then there exists a number c in (a. b) such that f(c)=N.

Definition of Horizontal Asymptote

The line y=L is called a horizontal asymptote of the curve y=f(x) if either:

the limit of f(x) as x approaches ∞ = L

or

the limit of f(x) as x approaches -∞ = L

Precise Definitions

Let f be a function defined on some interval (a, ∞)/(-∞. a) Then:

the limit of f(x) as x approaches ∞/-∞ = L

means that for every ε>0 there is a corresponding number N such that:

if x>/<N then |f(x)-L|<ε.

Tangent Line

The tangent line to the curve y=f(x) at the point p(a, f(a)) is the line through P with slope:

m=the limit of (f(x)-f(a))/(x-a) as x approaches a

or

the limit of (f(a+h)-f(a))/h as h approaches 0

provided that this limit exists.

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