P.o.D.
1.) Find the inclination in radians of the line 2x-7y+3=0.
2.) Find the angle in degrees between the lines 3x+2y-4=0 and
4x-y+6=0.
3.) Find the distance, in exact form, between the point (7,5) and the line y=5-x
1.) 0.2783 radians
2.) 47.7 degrees
3.)
7√2
2

12.1 – Introduction to Limits
Learning Target(s): I can use the definition of a limit to estimate limits; determine whether limits of a function exist; use properties of limits and direct substitution to evaluate limits.
EX: You are given 32 inches of wire and are asked to form a rectangle whose area is as large as possible. Determine the dimensions of the rectangle that will produce a maximum area.
Let w=width and L=length.
P=2w+2L  32=2w+2L
Solve for one of the variables.

32-2w=2L  16-w=L
We know that Area=L(W)
=(16-w)(w)
𝐴 = 16𝑤 − 𝑤 2
We can now graph this equation and find its maximum.
Max(8,64). This means that the maximum width is 8. We now need to find the corresponding length. L=16-w=16-8=8.
The dimensions are 8x8.
Estimating a Limit Numerically:

EX: Use a table to estimate numerically the limit: lim 𝑥→3
(5𝑥 −
3) .
X
2.9
2.99
2.999
F(x)
11.5
11.95
11.995
3.0
3.001
3.01
3.1
???
12.005
12.05
12.5
The limit appears to be 12.
EX: Use a table to estimate numerically the limit:

*We will use the TABLE feature on the graphing calculator.
The limit appears to be 6.
EX: Estimate the limit:
3
2
The limit appears to be 6.
Avoiding Common Errors:

Students often think of a limit as a quantity that can be approached but not reached.
Some limits are like that, but many limits are quantities that do exist on a function.
EX: Find the limit of f(x) as x approaches 5, where f is defined as 𝑓(𝑥) = {
1, 𝑥 ≠ 5
−2, 𝑥 = 5
.
*Draw the piecewise function on the whiteboard. f(5)=-2, but the limit of f(x) as x approaches 5 is 1. lim 𝑥→5
{
1, 𝑥 ≠ 5
−2, 𝑥 = 5
= 1

Limits That Fail to Exist:
EX: Show that the limit does not exist. lim 𝑥→0 𝑥
|𝑥|
*Let’s examine this one graphically.
Notice that we are approaching two different values (-1 or 1) depending upon which side of 0 we are looking. Since we do NOT approach the same value from both sides, no limit exists.
*We could also examine this on a
TABLE.

EX: Discuss the existence of the limit
4
*Let’s examine this graphically.
As x approaches 0 from either side, f(x) increases without bound (or to infinity). Because f(x) is NOT approaching a unique real number, you can

conclude that the limit does not exist. http://www.youtube.com/watc h?v=Lck5_YoxxGI
Oscillating Behavior:
EX: Discuss the example of the limit
*We want to examine this graphically.
In this graph, f(x) has oscillating

behavior between -1 and 1.
Therefore, the limit does NOT exist.
Basic Limits:
Let b and c be real numbers and let n be a positive integer.
1.
lim 𝑥→𝑐
2.
lim 𝑥→𝑐 𝑏 = 𝑏 a.
lim 𝑥→2
5 = 5 𝑥 = 𝑐
3.
a.
lim 𝑥→−3 lim 𝑥→𝑐 𝑥 𝑛 𝑥 = −3
= 𝑐 𝑛 a.
lim 𝑥→5 𝑥 2 = 5 2 = 25
4.
lim 𝑥→𝑐 𝑛
√𝑥 and c>0.
, where n is even

Evaluating Limits by Direct
Substitution:
EX: Find each limit: a.
lim 𝑥→2 b.
lim 𝑥→4 𝑥 3
8𝑥 c.
lim 𝑥→16
4
√𝑥 d.
lim 𝑥→
3𝜋
4
(tan 𝑥) a.) = 2 3 = 8 b.) = 8(4) = 32 c.) 4
= √16 = 2 d.) = tan (
3𝜋
4
) = −1
EX: Find each limit. a.) lim 𝑥→2
(𝑥 2 + 5𝑥 + 4)

b.) lim 𝑥→2
( 𝑥
2
+5𝑥+4 𝑥+4
) a.) = 2 2 + 5(2) + 4 = 4 + 10 + 4 = b.)
14 + 4 = 18
2
2
+5(2)+4
=
2+4
=
18
6
= 3
At the completion of this lesson, you should be able to:
1.
Define a limit, and find a limit using a table.
2.
Find a limit graphically.
3.
Determine when a limit does or does NOT exist.
4.
Evaluate limits by direct substitution.
For more information, visit http://teacherweb.com/TX/FriendswoodHig

hSchool/Nickerson/PAP-precal-12.1-day-1notes.pdf
HW Pg.860 3-54 3rds, 65-70
Worksheet 12-1