Page 143 - DMTH401_REAL ANALYSIS

Unit 10: Limit of a Function

Let a function f be defined for all values of x greater than a fixed number c. That is to say that f is Notes
defined for all sufficiently large values of x. Suppose that as x increases indefinitely, f(x) takes a
succession of values which approach more and more closely a value A. Further suppose that the
numerical difference between A and the values f(x) taken by the function can be made as small
as we please by taking values of x sufficiently large. Then we say f tends to the limit A as x tends
to infinity. More precisely, we have the following definition:
Definition 3: A function f tends to a limit A, as x tends to infinity if having chosen a positive
number , there exists a positive number k such that
|f(x – A)| >  " x  k.

The number E can be made as small as we like. Indeed, however small  we may take, we can
always find a number k for which the above inequality holds. We rewrite this definition in the
following way:

A function f(x)  A as x  w if for every  > 0, there exists k > 0 such that
|f(x) – A| <  for all x  k.
We write it as,
lim f(x) = A.
x 
This notion of the limit of a function needs a slight modification when x tends to –. This is as
follows:
We say that lim f(x) = A, if for a given  > 0, there exists a number k < 0 such that
x -
|f(x) – A| < E whenever x  k.
We write it as lim f(x) = A.
x -
Instead of f(x) approaching a real number A as x tends to + or –m, we may also have f(x)
Z
approaching + or – as x tends to a real number ‘a’. For example, if f(x) = l/x , x  0 and x takes
values near 0, the values of f(x) becomes larger and larger. Then we say that f(x) is tending to +
as x tends to 0. We can also have f(x) tending to +m or – as x tends to + or –. For example
f(x) = x tends to + or – as x tends to + or –. Again, the function f(x) = –x tends to + or –
as x tends to – or +. We formulate the following definition to cover all such cases of infinite
limits.
Definition 4: Infinite Limits of a Function
Suppose a is a real number. We say that a function f tends to +m when x tends to a, if for a given
positive real number M there exists a positive number  such that

f(x) > M whenever 0 < |x – a| < .
We write it as
lim f(x) = + .
-
x a
In this case we say that the function becomes unbounded and tends to + as x tends to a.
In the same way, f is said to – as x tends to a if for every real number –M, there is a positive
number  such that

f(x) < – M whenever 0 < |x – a| < .
We write it as
lim f(x) = –.
x a

LOVELY PROFESSIONAL UNIVERSITY 137