The Mathematics of Financial Modelingand Investment Management

4-PrincipCalculus Page 104 Friday, March 12, 2004 12:39 PM

104 The Mathematics of Financial Modeling and Investment Management

EXHIBIT 4.3 Most Common Definitions Associating the Relevant Condition to

Each Limit

The sequence tends to a finite

limit

The sequence tends to plus

infinity

The sequence tends to minus

infinity

Finite limit of a function

Finite left limit of a function

Finite right limit of a function

Finite limit of a function when

xtends to plus infinity

Finite limit of a function when

xtends to minus infinity

Infinite limit of a function

Infinite limit of a function

when xtends to plus infinity

lim a = a ∀ε > 0, ∃i(ε): |an – a| < ε

n ∞ → n for n> i(ε)

lim a = +∞ ∀D> 0, ∃i(D): a > D

n ∞ → n for n> i(ε) n

lim a = –∞ ∀D< 0, ∃i(D): an < D

n ∞ → n for n> i(ε)

lim fx() = d ∀ε > 0, ∃r(ε): |f(x) – d| < ε

x→ c

for |x– c| < r(ε)

lim fx() = d ∀ε > 0, ∃r(ε): |f(x) – d| < ε

x→ c – for |x– c| < r(ε), x< c

lim fx() = d ∀ε > 0, ∃r(ε): |f(x) – d| < ε

x→ c + for |x– c| < r(ε), x> c

lim fx() = d ∀ε > 0, ∃R(ε) > 0: |f(x) – a| < ε

x→ +∞ for x> R(ε)

lim fx() = d ∀ε > 0, ∃R(ε) > 0: |f(x) – a| < ε

x→ –∞

for x< –R(ε)

lim fx() = ∞ ∀D> 0, ∃r(D): |f(x)| > D

x→ c

for |x– c| < r(D)

lim fx() = +∞ ∀D> 0, ∃R(D): f(x) > D

x→ +∞

for x> r(D)

This definition does not imply that the function fis defined in an inter-

val; it requires only that cbe an accumulation point for the domain of

the function f.

A function can be right continuous or left continuous at a given

point if the value of the function at the point cis equal to its right or left

limit respectively. A function f that is right or left continuous at the

point ccan make a jump provided that its value coincides with one of

the two right or left limits. (See Exhibit 4.4.) A function y= f(x) defined

on an open interval (a,b) is said to be continuous on (a,b) if it is contin-

uous for all x∈ (a,b).

A function can be discontinuous at a given point for one of two rea-

sons: (1) either its value does not coincide with any of its limits at that

point or (2) the limits do not exist. For example, consider a function f

defined in the interval [0,1] that assumes the value 0 at all rational

points in that interval, and the value 1 at all other points. Such a func-