Paper 4: Fundamentals of Business Mathematics & Statistic

3.20 I FUNDAMENTALS OF BUSINESS MATHEMATICS AND STATISTICS

Calculus

Example 49: Discuss the continuity at x = 2 where

f(x) = 4x + 8, x ≠ 2

= 12 x = 2

Now (^) x 2lim f x lim 4 2 h 8 16→ +( )=h 0→ ( + )+ = (Putting x = 2 + h as x → 2, h → 0)
And (^) x 2lim f x lim 4 2 h 8 16.→ −( )=h 0→ ( − )+ = [for x = 2 – h, h → 0, as x → 2]
Again at x = 2, f(x) = f(2) = 12.
Three values are not equal. Hence f(x) is discontinuous at x = 2.
SELF EXAMINATION QUESTIONS

1. f (x) = x^2 + 1. Is the function continuous at x = 2? [Ans. Yes]


2. A functions is defined as follows f(x) = x^2 , x ≠ 2
   = 2, x = 2
   Is f(x) continuous at x = 2? [Ans. No]


3. Where are the following function discontinuous :

(i)^2 x (ii) x 1^1 − (iii)

2x^2

x 2− (iv)^2

(^1).
x 4x 3− + [Ans. (i) 0, (ii) 1, (iii) 2, (iv) 1, 3]

4. Discuss the continuity of f(x) at x = c, where

( )

x c^22

f x = x c−− x c≠

= 2c, x = c [Ans. Continuous]

5. (i) The function ( )
   x 4^2
   f x = x 2−− is undefined at x = 2. What value must be assigned to f(x), if f(x) is to be
   continuous at x = 2? Give reasons for your answer.

[Ans. ( )

x 4^2

f x = x 2−− , x 2 4,x 2≠ = = ]

(ii) ( )

x 9^2

f x = x 3−− when x ≠ 3. State the value of f(3) so that f(x) is continuous at x = 3.

[Ans. 6]

6. Is the function ( )
   x 8^3
   f x = x 2−− continuous at x = 2? If not what value must be assigned to f(2) to make f(x)
   continuous at x = 2. [Ans. 12]