Page 136 - DMTH401_REAL ANALYSIS

Real Analysis

Notes f(x) = 6, 5.8, 5.6, 5.4, 5.2, 5.02, 5.002
When x = .5, .6, .7, .8, .9, .99, .999
f(x) = 4, 4.2, 4.4, 4.6, 4.8, 4.98, 4.998

You can form a table for these values as follows:

X .5 .6 .7 .8 .9 .99 .999 1.001 1.01 1.1 1.2 1.3 1.4 1.5
f(x) 4 4.2 4.4 4.6 4.8 4.98 4.998 5.002 5.02 5.2 5.4 5.6 5.8 6

The limit of a function f at a point a is meaningful only if a is a limit point of its domain. That is,
the condition f(x)   as n  a would make sense only when there does not exists a nbd. U of a
for which the set U n Dom (f)\{a} is empty i.e., a  (Dom 0))’.

You see that as the values of x approach 1, the values of f(x) approach 5. This is expressed by
saying that limit of f(x) is 5 as x approaches 1. You may note that when we consider the limit of
f(x) as x approaches 1, we do not consider the value of f(x) at x = 1.

Thus, in general, we can say as follows:
Let f be a real function defined in a neighbourhood of a point x = a except possibly at a. Suppose
that as x approaches a, the values taken by f approach more and more closely a value A. In other
words, suppose that the numerical difference between A and the values taken by f can be made
as small as we please by taking values of x sufficiently close to a. Then we say that f tends to the
limit A as x tends to a. We write

f(x)  A as x  a or lim f(x) = A.
x a
This intuitive idea of the limit of a function can be expressed mathematically as formulated by
the German mathematician Karl Veierstrass in the 18th Century. Thus, we have the following
definition:

Definition 1: Limit of a Function
Let a function f be defined in a neighbourhood of a point ‘a’ except possibly at ‘a’. The function
f is said to tend to or approach a number A as x tends to or approaches a number ‘a’ if for any
 > 0 > there exists a number  > 0 such that
|f(x) – A| <  for 0 < |x – a| < 6.

We write it as lim f(x) = A. You may note that
x a
|f(x) – A| <  for 0 < |x – a| < 6.
can be equivalently written as

f(x) ]A – , A +  [ for x ] a – 6, a +  [ and  a.
Geometrically, the above definition says that, for strip S of any given width around the point A,
A
if it is possible to find a strip S of some width around the point a such that the values that f(x)
a
takes, for x in the strip S (x  a), lies in the shaded box formed by the intersection of strips S and
a A
S , then lim f(x) = A.
a
x a
This is shown geometrically in Figure 10.1 below. The inequality 0 < |x – a| < 6 determines the
interval ] a – 6, a + [ minus the point ‘a’ along the x-axis and the inequality |f(x) – A| < 
determines the interval ]A – , A + [ along the y-axis.

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