Page 151 - DMTH401_REAL ANALYSIS

Unit 10: Limit of a Function

4. The function f is said to tend to a number A as x tends to ‘a’ from the left or through values Notes
smaller than ‘a’ if given a number E > 0, there exists a number  > 0 such that ..............
.....................................

10.4 Summary

 We started with the intuitive idea of a limit of a function. Then we derived the rigorous
definition of the limit of a function, popularly called  –  definition of a limit. Further,
we gave the notion of right and left hand limits of a function. It has been proved that
lim f(x) = A if and only if both right hand and left hand limits are equal to A i.e. lim f(x)
a
x a x +
= lim f(x) = A. In the same section we discussed the limit of a function as x tends to + or
x a–
–. Also we discussed the infinite limit of a function.
 We studied the idea of sequential limit of a function by connecting the idea of limit of an
arbitrary function with the limit of a sequence. It has been shown how this relationship
helps in finding the limits of functions.

 We defined the algebraic operations of sum, difference, product, quotient of two functions.
We proved that the limit of the sum, difference, product and quotient of two functions at
a point is equal to the sum, difference, product and quotient of the limits of the functions
at the point provided in the case of quotient, the limit of the function in the denominator
is non-zero. Finally in the same section, the usefulness of the algebra of limits in finding
the limits of complicated functions has been illustrated.

10.5 Keywords

Function: 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.
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| < .

10.6 Review Questions

2
x - x 18
+
1. Show that Lim = 4, using the  –  definition.
-
x 2 3x 1
2. Find the limit of the function f defined as
2
2x + x
f(x) = , x  0 when x tends to 0.
3x
3. Find, if possible, the limit of the following functions.
-
x 2
(i) f(x) = , x 2
x 2
-
when x tends to 2.
- 1
(ii) f(x) = 1 x , x  0
e + 1
when x tends to 0.

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