§ 1-4 Limits and Continuity
The student will learn about:
limits, finding limits,
one-sided limits,
infinite limits,
and continuity.
1
Limits
The word “limit” is used in everyday
conversation to describe the ultimate behavior
of something, as in the “limit of one’s
endurance” or the “limit of one’s patience.”
In mathematics, the word “limit” has a similar
but more precise meaning.
Limits
Given a function f(x), if x approaching 3 causes
the function to take values approaching (or
equalling) some particular number, such as 10,
then we will call 10 the limit of the function and
write
In practice, the two simplest ways we can
approach 3 are from the left or from the right.
Limits
For example, the numbers 2.9, 2.99, 2.999, ...
approach 3 from the left, which we denote by
x→3 –, and the numbers 3.1, 3.01, 3.001, ...
approach 3 from the right, denoted by x→3 +.
Such limits are called one-sided limits.
Example 1 – FINDING A LIMIT BY TABLES
Use tables to find
Solution :
We make two tables, as shown below, one with x
approaching 3 from the left, and the other with x
approaching 3 from the right.
Limits IMPORTANT!
This table shows what f (x) is doing as x approaches 3.
Or we have the limit of the function as x approaches
We write this procedure with the following notation.
lim 2x  4  10
x3
Def: We write
10
lim f (x)  L
3
x c
or as x → c, then f (x) → L
if the functional value of f (x) is close to the single real number L
H
whenever x is close to, but not equal to, c. (on either side of c).
x
2
2.9
2.99
2.999
3
3.001
3.01
3.1
4
f (x)
8
9.8
9.98
9.998
?
10.002
10.02
10.2
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Limits
As you have just seen the good news is that
many limits can be evaluated by direct
substitution.
Limit Properties
These rules, which may be proved from the
definition of limit, can be summarized as
follows.
For functions composed of addition,
subtraction, multiplication, division, powers,
root, limits may be evaluated by direct
substitution, provided that the resulting
expression is defined.
lim
f (x)  f (c)
xc
8
Examples – FINDING LIMITS BY
DIRECT SUBSTITUTION
1.
lim
4 2
Substitute 4 for x.
62
36
x2


4
x3 63 9
Substitute 6 for x.
x
x4
2.
lim
x6
Example
But be careful when a quotient is involved.
x2  x  6 0
lim

x2
x2
0
Graph it.
Which is undefined!
But the limit exist!!!!
What happens at x = 2?
x2  x  6
(x  3)(x  2)
lim
 lim
 lim (x  3)  5
x2
x2
x2
x2
x2
x2  x  6
NOTE : f ( x ) 
graphs as a straight line.
x2
10
One-Sided Limit
We have introduced the idea of one-sided
limits. We write
lim f ( x)  K

xc
and call K the limit from the left (or lefthand limit) if f (x) is close to K whenever x is
close to c, but to the left of c on the real
number line.
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5
One-Sided Limit
We write
lim f ( x)  L
x  c
and call L the limit from the right (or righthand limit) if f (x) is close to L whenever x is
close to c, but to the right of c on the real
number line.
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The Limit
Thus we have a left-sided limit:
And a right-sided limit:
lim f ( x)  K

xc
lim f ( x)  L
x  c
And in order for a limit to exist, the limit
from the left and the limit from the right
must exist and be equal.
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Example
f (x) = |x|/x at x = 0
lim
x
 1
x
lim
x
 1
x
x0
x0
0
The left and right limits are different, therefore
there is no limit.
14
Infinite Limits
Sometimes as x approaches
c, f (x) approaches infinity
or negative infinity.
Consider
lim
x2
1
 x  2
2
From the graph to the right you can see that the limit
is ∞. To say that a limit exist means that the limit is a
real number, and since ∞ and - ∞ are not real numbers
means that the limit does not exist.
15
Intro to Continuity
As we have seen some graphs have holes in
them, some have breaks and some have
other irregularities. We wish to study each
of these oddities.
We will use our
information of limits
to decide if a function
is continuous or has
holes.
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Continuity
Intuitively, a function is said to be continuous
if we can draw a graph of the function with
one continuous line. I. e. without removing
our pencil from the graph paper.
Definition
A function f is continuous at a point x = c if
1.
f (c) is defined
2.
lim f(x) exists
3.
lim f(x)  f (c)
x c
x c
THIS IS THE DEFINITION OF
CONTINUITY
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Example
f (x) = x – 1 at x = 2.
a. f (2) = 1
1
x11
b. xlim
2
The limit
exist!
c. f (2)  1  lim x  1
2
x2
Therefore the function is continuous at x = 2.
19
Example
f (x) = (x2 – 9)/(x + 3) at x = -3
a. f (-3) = 0/0
b.
Is undefined!
x2  9
lim

x  3 x  3
-6
-3
The limit exist!
c.
x2  9
lim
 f ( 3)
x  3 x  3
-6
Therefore the function is not
continuous at x = -3.
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You can use table on your calculator to verify this.
Continuity Properties
If two functions are continuous on the
same interval, then their sum, difference,
product, and quotient are continuous on
the same interval except for values of x
that make the denominator 0.
Every polynomial function is continuous.
Every rational function is continuous
except where the denominator is zero.
21
Continuity Summary.
Functions have three types of discontinuity.
2
Consider x  4x  5
f (x) 
x2  2x  15
Graph on your calculator
with a standard window.
1. Discontinuity at
vertical asymptote.
2. Discontinuity at hole.
3. We have discontinuity with some functions
that have a gap.
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Summary.
• We learned about limits and their properties.
• We learned about left and right limits.
• We learned about continuity and the properties of
continuity.
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ASSIGNMENT
§1.4 On my website.
20, 21.
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