I. The Limit Laws
Math131 Calculus I
The Limit Laws
Notes 2.3
I. The Limit Laws
Assumptions: c is a constant and lim f (x) and lim g(x) exist
x a
xa
Limit Law in symbols
1
lim[ f (x) + g(x)] = lim f (x) + lim g(x)
xa
xa
xa
2
lim[ f (x) - g(x)] = lim f (x) - lim g(x)
xa
xa
xa
3
lim cf (x) = c lim f (x)
xa
xa
4
lim[ f (x)g(x)] = lim f (x) lim g(x)]
xa
xa
xa
5
lim
f
(x)
=
lim
xa
f
(x)
xa g(x) lim g(x)
( ) if lim g(x) 0 x a
xa
6
lim[ f (x)]n = [lim f (x)]n
xa
xa
7
lim c = c
xa
8
lim x = a
xa
9
lim x n = a n
xa
10
lim n x = n a
xa
11
lim n f (x) = n lim f (x)
xa
xa
Limit Law in words The limit of a sum is equal to
the sum of the limits.
The limit of a difference is equal to the difference of the limits.
The limit of a constant times a function is equal to the constant times the limit of the function.
The limit of a product is equal to the product of the limits.
The limit of a quotient is equal to the quotient of the limits.
where n is a positive integer
The limit of a constant function is equal to the constant.
The limit of a linear function is equal to the number x is approaching.
where n is a positive integer
where n is a positive integer & if n is even, we assume that a > 0
where n is a positive integer & if n is even, we assume that lim f (x) > 0
xa
Direct Substitution Property:
If f is a polynomial or rational function and a is in the domain of f, then lim f (x) =
x a
"Simpler Function Property":
If f (x) = g(x) when x a then lim f (x) = lim g(x) , as long as the
xa
xa
limit exists.
Math131 Calculus I
Notes 2.3
page 2
ex#1 Given lim f (x) = 2 , lim g(x) = -1, lim h(x) = 3 use the Limit Laws find lim f (x)h(x) - x2 g(x)
x3
x3
x 3
x3
ex#2 Evaluate lim 2x2 + 1 , if it exists, by using the Limit Laws.
x2 x2 + 6x - 4
ex#3 Evaluate:
lim 2x 2 + 3x - 5
x 1
ex#4 Evaluate:
lim1- (1- x)2
x0
x
ex#5 Evaluate:
lim h + 4 - 2
h0
h
Math131 Calculus I
Notes 2.3
page 3
Two Interesting Functions
1. Absolute Value Function
Definition:
x
=
-
x x
if if
x0 x ................
................
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