Paper 4: Fundamentals of Business Mathematics & Statistic

3.16 I FUNDAMENTALS OF BUSINESS MATHEMATICS AND STATISTICS

Calculus

Example 42 :

x x

x 0

lima b

→ x

−

Solution:

Expression

x x

x 0

lima 1 b 1

→ x

= − − + x x

x 0 x 0

lima 1 b 1lim

→ x → x

= − − −

= log a – log b [by (D) above] = log ba.

Example 43 :

ax bx

x 0

lime e

→ x

−

Solution:

Expression ( ) ( )

ax bx ax bx

x 0 x 0 x 0

e 1 e 1 e 1 e 1

lim→ x lim→ x lim→ x

− − − − −

= = −

u v

u 0 v 0

lime 1 e 1lim

→ u/a → v /b

= − − −

u

u 0 0

a lime 1 blime 1

u

ν

→ ν →

= − − −

ν

= a. 1 – b.1 = a – b.

SELF EXAMINATION QUESTION

Show that :

1. (i)

2

x 1

limx 1 2

→ x 1

− =

− (ii)

2

x 2

limx 4 4

→ x 2

− =

− (iii)

2

x 2

lim x 4 4

→ − x 2

− = −

+

2. (i) xlim→ (^2) x 13x+ [Ans. 2] (ii) x 3lim x 1 x 2→( +)( + ) [Ans. 20]
   (iii) x 0lim x 3x 7→ (^2 + − ) [Ans.–7] (iv)
   2
   x 0
   limh 4h
   → 2h

• [Ans. 2]
  (v)
  2
  x 2
  lim3x 4x 7
  → 3x 5

− +

− [Ans. 11] (vi)

( )^2

x 2

lim2x 1

→ x 1

−

+ [Ans. 3]

3. (i) xlim→ 0 1 x 1 x+ x− − [Ans.1] (ii)

2

x 0

lim 1 x 1

→ x

+ − [Ans. 0]

(iii)

2

x 0^2

lim1 1 x

→ x

− − [Ans. 1

2 ] (iv)^ x^0

lim 1 2x 1 3x

→ x

+ − − [Ans. 5

2 ]

Do the following limits exists? If so find the values.

4. (i) x 3limf x ,→ ( ) when f (x) = 2x + 3, x > 3
   = 3x + 1, x £ 3. [Ans. doesn’t exist]