26/09/2014
f(x) = function of x, f(x) = x2 Р9 / x Р3, f(y) = function of y , f(y) = y2 Р9 / y Р3 ̬ y = f(x)
Limits of function: In mathematics, a limit is the value that a function or sequence "approaches" as the input or index approaches some value. Limits are essential to calculus (and mathematical analysis in general) and are used to define continuity, derivatives, and integrals.
Limit existence: when L.H.L = R.H.L
Sign of “limit of function”: lim(x à a) f(x) ¤lim(x à a) f(x) = M or lim(x à a) f(x) = (x2 + x + M)
M = real no. and real no consist on basic types of no.
v Positive integer: 1, 2, 3, 4, 5, 6, 7, 8 . . . ∞
v Negative integer: -1, -2, -3, -4, -5, -6 . . . ∞
v Whole No: 0, 1, 2, 3, 4, 5, 7, 8 . . . ∞
v Rational No: 2/3, 3/5, 5/7, 4/9 . . . ∞
v Irrational No: 
, 
, 
, 
, 
. . . ∞
Example:
1) lim(x à 2) f(x), lim(x à 2) = (x3-1)/x2 à {(2)3- 1}/(2)2 à (8 – 1) / 4 à 7/4 ANS
2) lim(x à 2) f(x), lim(x à 2) = 5 à 5 ANS
Left hand limit:if constant “a” power = - like a- | Right hand limit: if constant “a” power = + like a+ | ||
| Example: f(x) = { | 20x ; x < 4 X2 + 5 ; x > 4 | } find left & right hand limit | |
| Left hand limit = lim(x à a-) f(x) | Right hand limit = lim(x à a+) f(x) | ||
| lim(x à 4-) = 20x = 20(4) = 80 | lim(x à 4+) = x2 + 5 = (4)2 + 5 = 16 + 5 = 21 | ||
| Existence = L.H.L = R.H.L but 80 ≠ 21 so limit not exist | |||
Limits of infinity: lim(x à ∞) = f(x)
Note:
1) answer can be in “M” or ∞
2) 0/0 or ∞/∞ = no answer
Example: Page no 677 & exp no 18: lim(x à ∞) = (3x2 + 5x)/ (4x2- 5), according to 2) we must simplify it 1st.
lim(x à ∞) = x2 ( 3 + 5/x) / x2 (4 - 5/x2) à lim(x à ∞) = ( 3 + 5/x) /(4 - 5/x2) à = ( 3 + 5/∞) /(4 - 5/∞2)
= (3 + 0) / (4 – 0) Ã = 3/ 4 Answer: ¾
Home Assignment: Page 670 (5 to 8 Find limit, 7 to 10 limit existence, 11 to 28 solve)
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