Paper 4: Fundamentals of Business Mathematics & Statistic

3.14 I FUNDAMENTALS OF BUSINESS MATHEMATICS AND STATISTICS

Calculus

Example 36 : Find

2

x^2

lim 5 2x.

→ ∞3x 5x

−

+

Solution:

Expression

2 2

x u 0

(^52)
limx lim5u 2
(^35) 3u 5
x
→ ∞ →

− −

= = +

+

(^1) u, asx ,u 0
x
(^) = → ∞ →
(^)
2
u 0
u 0
5 lim u 2 0 2 2
3 lim u 5 0 5 5→.
→

− −

= + = + = −

Problems regarding rationalisation :

Example 37 : Find the value of : hlim→ 0 x h x+h−

( )( )

h 0 h 0 ( ) h 0 ( )

x h x x h x x h x x h x

lim h lim lim

→ → h x h x → h x h x

+ − + − + + + −

= =

+ + + +

h (^0) ( ) h 0
lim h lim^1 ,
→ h x h x → x h x
= =

• 
  
  
  
  • 
    
    
    
    • 
      
      
      
      • (h ≠ 0)
        (^11).
        x 0 x 2 x
      
      
      
      
    
    
    
    
  
  
  
  

= =

+ +

Problem regarding both sides limits :

Example 38 : A function is defined as

f(x) = x^2 , for x > 1

= 4.1, for x = 1

= 2x, for x < 1.

Does lim f(x)x 1→ exist?

Solution:

Let us find x 1lim f(x)→ + and x 1lim f(x).→ −

Now lim f(x) lim x 1 1,x 1→ + =x 1→ + 2 2= = as f(x) = x^2 for x > 1

Again x 1lim f(x) lim 2x 2.1 2,→ − =x 1→ − = = as f(x) = 2x for x < 1

Since the two limiting values are not equal, so lim f xx 1→ ( ) doesn’t exist.