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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