FUNDAMENTALS OF BUSINESS MATHEMATICS AND STATISTICS I 3.7]c , c [ , 0− ∈ + ∈ ∈ >Is the neighbourhood of c.
Any neighbourhood from which the point c is excluded is called deleted neighbourhood of c.
Geometrically it means set of those points which are within an ∈ distance from c on either side except for
the point c.
(iv) Limit of a function f (x). A number l is said to be the limit of f(x) as x→a written as
x a
Lt f(x)
→
= l if
a. The function is defined and single valued in the deleted neighbourhood of a.
b. For every positive number ∈, however small, there exists a positive number δ (usually depending
on ∈), such
|f(x) 1|− < ∈
Whenever 0 |x a|< − < δ
From the above definition it is interesting to note that Ltx a→ f(x) may exist, even if the function f(x) is
not defined at x = a. Sometimes both the things may happen, i.e.,
(i) The function is defined at x = a, and
(ii)
x a
Lt f(x)
→
also exists.Right hand and left hand limits
The variable point x can approach a either from the left or from the right. These respective approaches are
indicated by writing
x→a–
and x→a+
If x aLt→ −f(x) = l 1 (left hand limit)
and x aLt→ +f(x) = l 2 (right hand limit)
we say x aLt→ − f(x) = l if and only if
l 1 = l 2 = l
Methods of finding limit of a function f(x) as x tends to a finite quantity say ‘a’
There are three methods for finding limit of a function f(x) as x tends to a finite quantity say ‘a’ :
(i) Method of factors