Warm Up: No Calc1. Find all asymptotes for

(A) x=1, x=-1, y=1 (B) x=1, y=1 (C) x=1, x=-1, y=0 (D) x=1, x=-1 (E) y=1

2.

3. Use properties of logarithms to decide which of the following is largest.

2

2( )

1

x xf x

x

ln 4( ) ln(30) ln(2) ( )2 ln 4 ( ) ln(3) ln(4) ( )

ln 2A B C D

Pick up new packet!

If we increase the number of sides of the polygon, what can you say about the polygon

with respect to the circle?

What is a limit?

Limit is the value of Y as X approaches a given #:

Lxfcx

)(lim

3 Kinds of Limits:

Left – Hand Limit

As x approaches from the left side of c

Right – Hand Limit

As x approaches from the right side of c.

Limit (double – sided)

As X approaches c from either direction.

Only exists if left-hand and right-hand are the same.

Lxfcx

)(limLxfcx

)(limLxfcx

)(lim

When do limits not exist? DNE

Video

THM: Existence of a Limit

lim

lim

lim

x c

x c

x c

f x L iff

f x L AND

f x L

Example 1: Find the following limits

Practice

1

1-

1

0

0

0-

0

-3

-2

-2-

-2

2

Grab a graphing board, marker, and towel

Limit Properties

These are important!

Limits to KnowLet b & c be real numbers and let n be a positive integer.

1. The limit of a constant function is the constant.

2. The limit at any x-value on the line y=x IS the x-value itself.

3. The limit at any x-value of any function of the form y = xn is the x-value raised to the nth power.

limx cb b

limx cx c

limnn

x cx c

Practice:

11

7

3

2

1. lim 8

2. lim

3. lim

x

x

x

x

x

Properties of LimitsLet b & c be real # and n a positive int. and

lim ( ) lim ( )x c x cf x L g x K

1.lim[ * ] *x cb f x b L

2.lim[ ]x c

f x g x L K

Scalar multiple

Sum or Differ.

3.limx c

f x g x LK

4.lim , 0x c

f x LK

g x K

Product

Quotient

5.lim[ ]n n

x cf x L

Power

Practice1.

2.

Another nice thing about limits…• They help us find holes in the graph.• Ex: What will happen at x=1?

1,1

1)(

3

xx

xxf

x .75 .9 .99 .999 1 1.001 1.01 1.1 1.25

f(x) ?

1,1

1)(

3

xx

xxf

While f(1) D.N.E., as x moves arbitrarily close to 1 from the left and right, f(x)

moves arbitrarily close to 3.

“The limit of f(x) as x approaches 1 is 3”

3)(lim1

xfx

Example: Graph )3)(2(

2

xx

xy

Homework:

pg. 65 (1 – 4, 37 – 48, 79-82)Packet pg. 1