The Mathematics of Financial Modelingand Investment Management

4-PrincipCalculus Page 102 Friday, March 12, 2004 12:39 PM

102 The Mathematics of Financial Modeling and Investment Management

LIMITS

The notion of limit is fundamental in calculus. It applies to both func-

tions and sequences. Consider an infinite sequence S of real numbers

S ≡ (a 1 , a 2 , ..., ai,...)

If, given any real number ε > 0, it is always possible to find a natural

number i(ε ) such that

ii≥ () ε impliesai – a ε <

then we write

lim a = a

n → ∞ n

and say that the sequence S tends to a when n tends to infinity, or that a

is the limit of the sequence S.

Two aspects of this definition should be noted. First, ε can be chosen

arbitrarily small. Second, for every choice of ε the difference, in absolute

value, between the elements of the sequence S and the limit a is smaller

than ε for every index i above i(ε ). This translates the notion that the

sequence S gets arbitrarily close to a as the index i grows.

We can now define the concept of limit for functions. Suppose that a

real function y = f(x) is defined over an open interval (a,b), i.e., an inter-

val that excludes its end points. If, given any real number ε > 0, it is

always possible to find a positive real number r(ε ) such that

xc– <r() ε implies yd– ε <

then we write

lim fx() = d

x → c

and say that the function f tends to the limit d when x tends to c.

These basic definitions can be easily modified to cover all possible

cases of limits: infinite limits, limits from the left or from the right or

finite limits when the variable tends to infinity. Exhibit 4.2 presents in

graphical form these cases. Exhibit 4.3 lists the most common defini-

tions, associating the relevant condition to each limit.