Transcript § 1-1 Functions

Limits and Continuity
The student will learn about:
limits, finding limits,
one-sided limits,
infinite limits,
and continuity.
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Simple definition:
A limit is the intended height
of a function.
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THE LIMIT (L) OF A FUNCTION IS
THE VALUE THE FUNCTION (y)
APPROACHES AS THE VALUE OF
(x) APPROACHES A GIVEN VALUE.
lim f ( x)  L
x a
When does a limit exist?
• A limit exists if you travel along a function
from the left side and from the right side,
towards some specific value of x, as long as
the function meets at the same height.
• A general limit exists on f(x) when x = c if
the right and left hand limits are both equal
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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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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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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.
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Three methods of evaluating
limits
• Substitution
• Factoring
• The Conjugate Method
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Solving Limits using direct
substitution
•Direct substitution is the easiest
way to solve a limit
•Can’t use it if it gives an undefined
answer
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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
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Examples – FINDING LIMITS BY
DIRECT SUBSTITUTION
1.
lim
4 2
Substitute 4 for x.
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x2


4
x3 63 9
Substitute 6 for x.
x
x4
2.
lim
x6
Example 1 – FINDING A LIMIT BY
SUBSTITUTION AND 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
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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.
Factoring method
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
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Using direct substitution is not always as
evident. Find the limit below.
x  2x  3
lim
x 1
x 1
2
Rewrite before substituting
Factor and cancel common factors –
then do direct substitution.
The answer is 4.
The conjugate Method
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Limit and Infinity
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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.
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Limits at Infinity (horizontal)
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Conclusion
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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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What makes a function continuous
Continuous functions have:
– No breaks in the graph
– No jumps
– No holes
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Types of discontinuity
Point or hole
infinite or vertical jump or gap
asymptote
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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.
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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.
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Continuity Summary.
Is the function continuity? If not what type of
discontinuity is it.
x  4x  5
f (x)  2
x  2x  15
2
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Examples
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