3.8 I FUNDAMENTALS OF BUSINESS MATHEMATICS AND STATISTICS

Calculus

(ii) Method of substitution

(iii) Method of rationalization.

In method I, if f(x) is of the form

g(x)

h(x) factorise g(x) and h(x), cancel the common factors and then put the

value of x.

Example 24: Find the value of

3

1 2

1

→ 1

−

x −

Lt x

x.

Solution:

Now,

3 2

2

1 ( 1)( 1)

1 ( 1)( 1)

− = − + +

− − +

x x x x

x x x

(^21)
1

= + +

+

x x

x

∴

3 2

1 2 1

Lt^1 Lt^1

→ 1 → 1

− = + +

x − x +

x x x

x x

1 1 1 3

1 1 2

= + + =

+.

Method II

The following steps are involve:

(i) Put x = a + h where h is very small but ≠ 0, i.e. x→ a, h→0.

(ii) Simplify numerator and denominator and cancel common powers of h.

(iii) Put h = 0.

The result is the required limit.

Example 25: Evaluate Lt→ −−

n n

x a

x a

x a.

Solution:

Put x = a + h where h is very small, then

− =( )+ −

−

x a a h an n n n

x a h

1 1

+ −

=^

n

an h

a

h

2

1 ( 1) 2 1

(^) + + − + (^) −
=^
an nha n n ha L
h
2
( 1) 2
(^) + − +
(^)

a n n nn ah ha L
h