Paper 4: Fundamentals of Business Mathematics & Statistic

3.12 I FUNDAMENTALS OF BUSINESS MATHEMATICS AND STATISTICS

Calculus

(i) f(a) does not exist, but x alim f x→ ( ) exists.

(ii) f(a) exists, but x alim f x→ ( ) does not exist.

(iii) f(a) and x alim f x→ ( ) both exist, but unequal.

Let f(x) = 0, for x ¹ 0

= 1, for x = 0

x 0lim f x 0 lim f(x)→ + ( )= =x 0→ − and f(0) = 1, unequal values.

(iv) f (a) and x alimf(x)→ both exist and equal.

(v) neither f(a) or, x alimf(x)→ exists.

Fundamental Theorem on Limits :

If x alimf(x)→ = l and x alim (x) m,→φ = where l and m are finite quantities then

1. x alim f x x→ ( )± φ( )^ = l ± m


2. lim f(x). xx a→^ φ( )^ = lm.

3.

( )

x a ( )

limf x l,m 0

→ φ =mx ≠

4. If x alim x b→ φ( )= and x blimf(u) f(b)then→ =

x blim f x f lim x f(b).→ {φ( )}= {x a→ φ( )}=

Example 32 : Evaluate,

2

x 1

limx 3x 1.

→ 2x 4

+ −

+

Solution:

As the limit of the denominator ≠ 0, we get

Expression

( )

( )

2 2

x 1 x 1 x 1 x 1

x 1 x 1 x 1

lim x 3x 1 limx lim3x lim1

lim 2x 4 lim 2x lim4

→ → → →

→ → →

+ − + −

= + = + (by therom 1)

1 3.1 1 3 1

2.1 4 6 2

= + − = =

+

We have not applied the definition to save labour. If we substitute x = 1, we get the value of the function

1

= 2 (equal to the limit the value as x →1). Practically this may not happen always, as shown below.