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Evan B.

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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

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

The Intermediate Value Theorem

Definition of Horizontal Asymptote

Precise Definitions

Tangent Line

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