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
- x alim f x x→ ( )± φ( )^ = l ± m
- lim f(x). xx a→^ φ( )^ = lm.
3.
( )
x a ( )
limf x l,m 0
→ φ =mx ≠
- 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.