Calculus Chapter One
Sec 1.2
Limits and Their Properties
Limits and Their Properties

What is a “limit”?

What if someone says you are “pushing
them to the limit”?

Usually (in English) the word limit is used
to mean a boundary beyond which one
cannot go.
Limits and Their Properties

Think about a game warden who catches
a hunter. The hunter might say he
“caught the limit” or “shot the limit”.

The number can approach the limit and
even reach it, but it cannot exceed it.
Limits and Their Properties

In math, a limit is the number that a
function approaches as the values of x
plugged into the function approach a
fixed number.
Limits and Their Properties

For example, suppose you move your
nose toward a fan. Imagine your nose at
position x and the fan at position 3, 3 feet
from the origin.

Draw example.
Limits and Their Properties

We want to know what happens as x get
really close to 3.

What happens are your nose
approaches 3, getting closer and closer,
without ever really reaching 3?
Limits and Their Properties

You feel the breeze stronger as x gets
closer to 3.

We want to know what happens to the
amount of the breeze as you approach 3.
Limits and Their Properties

We are taking lim b(x), where b(x)
X→3
is the breeze that you feel when your
nose is at point x.
Limits and Their Properties

Say you feel a breeze of 6 mph when
x=2.9 and the breeze increases as you
move your nose toward the fan as in the
chart:
Nose Position
Breeze
2.9 2.99
2.9999 2.9999
2.99999
6
6.92
6.999993
6.7
6.991
Limits and Their Properties

It looks like the breeze is approaching 7
mph as your nose approaches the fan.

So, we say that the lim b(x)=7
x→3
“The limit of b of x as x approaches 3 is 7.”
Limits and Their Properties

Example 1: lim x^2-7x^3+5
x→1
Substitute x=1
Answer = -1
Limits and Their Properties

Example 2: lim
x→2

1
(x-2)^2
x≠2 so try values closer and closer to 2
if x=0→1/4
if x=1→1
if x=1.5→4
if x=1.75→16
if x=1.95→400
if x=1.9567→533.36
Limits and Their Properties
As x approaches 2, (x-2)^2 gets very
small.
 But 1 divided by a TINY number is a very
LARGE number.
 So, as x approaches 2, 1/(x-2)^2
approaches infinity (∞).
 Since ∞ is not a real number, we say that
the limit does not exist.

Limits and Their Properties

Example 2 shows that we can have
limits that are so big (bigger than any
real number) that the limit does not exist.

We can have limits that don’t exist for
other reasons.
Limits and Their Properties

Example 3 lim sin(1/x)
x→0
 As
x gets smaller and smaller, 1/x gets
bigger and bigger. But the sin (1/x) will
always be between 1 and -1, since the
sine of any number is between 1 and -1.
Limits and Their Properties
 As
1/x gets bigger and bigger, sin (1/x)
oscillates between 1 and -1 faster and
faster.
 It goes crazy and is not getting closer
and closer to any one number.
 It does not zero in on any one value.
 Therefore, it has NO LIMIT!!
Limits and Their Properties

Think of it as someone who is “falling in
love” and can’t commit. They want to
play the field….a perpetual swinger!!!

See graph on page 50.
Limits and Their Properties

Example 4 (Hard limits can be easy at
other points.)
lim sin (1/x)
x→2/π
 Answer:
1
 The limit of sin (1/x) has problems only
as x→0, not when x approaches other
values.
Limits and Their Properties
General Procedures for Taking a Limit

We want to take lim f(x) for some
x→b
function f(x).
Limits and Their Properties

Always plug b into the function. If you
get a number (that doesn’t have 0 in the
denominator or a negative number inside
a square root) and if the function is not
one of those weird functions that
changes its definition at the point B, then
the number f(b) is the limit.
Limits and Their Properties
“Plugging in” always works for a
polynomial. It usually works for almost
any function as long as the point you are
plugging in doesn’t have you divide by
zero.
 This is called DIRECT SUBSTITUTION.

Limits and Their Properties
Examples 5-6
 Examples 7-8
 No problem with 0 in the numerator.
Worry if it is in the denominator.
 If 0 is in the denominator…try to simplify
the fraction, hoping to get rid of one of
the zeros.
 Examples 9 and 10

Limits and Their Properties

Classwork (page 53)
#4, 6, 8, 10, 12, 14, 16, 18, 20

Homework (page 53)
#3, 5, 7, 9, 11, 13, 15, 17, 19