Limits
Chapter 1 & 2Prof. Ibrahim El-Henawy
Note that
Remember that the symbol Σ (sigma) represents
“the sum of.”
ما معنى االتي
كميه معينه
كميه غير معينة
كميه غير معرفة
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.
Use tables to find
Example– FINDING A LIMIT BY TABLES
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.
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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.
x 2 2.9 2.99 2.99
9
3 3.00
1
3.01 3.1 4
f (x) 8 9.8 9.98 9.99
8
? 10.002 10.02 10.2 12
x 3lim 2x 4 10
Def: We write
if the functional value of f (x) is close to the single real
number L whenever x is close to, but not equal to, c. (on
either side of c).
or as x → c, then f (x) → L
3
10
x clim f (x) L
Limits
As you have just seen the good news is that many limits can be evaluated by direct substitution.
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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.
cx
f (x) )cf (lim
Examples – FINDING LIMITS BY DIRECT SUBSTITUTION
x 4
1. xlim
Substitute 4 for x.4 2
2
x 6
x2.
x 3lim
26 36
46 3 9
Substitute 6 for x.
DirectSubstitution
DirectSubstitution
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Direct Substitution
But be careful when a quotient is involved.
2
x 2
x x 6 0lim Which is undefined!
x 2 0
2
x 2 x 2 x 2
x x 6 (x 3)(x 2)lim lim lim (x 3) 5
x 2 x 2
2x x 6
NOTE : f ( x ) graphs as a straight line.x 2
Graph it.
But the limit exist!!!!
What happens at x = 2?
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One-Sided Limit
We have introduced the idea of one-sided
limits. We write
and call K the limit from the left (or left-
hand 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.
K)x(flimcx
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One-Sided Limit
We write
and call L the limit from the right (or right-
hand 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.
L)x(flimcx
The Limit
K)x(flimcx
Thus we have a left-sided limit:
L)x(flimcx
And a right-sided limit:
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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Examplef (x) = |x|/x at x = 0
x 0
xlim 1
x
The left and right limits are different, therefore there is no
limit.
0x 0
xlim 1
x
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Infinite Limits
Sometimes as x
approaches c, f (x)
approaches infinity or
negative infinity.
2
x 2
1lim
x 2 Consider
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.
IndeterminateForms ∞/∞, -∞/ ∞, 0/0
Dealing withIndeterminate Forms
Factor and Reduce
T# 1
Divide by Largest Power of the Variable
T#2
Use the Common Denominator
Rationalize the Numerator (or Denominator)
RATIONALIZE THE DENOMINATOR
Limits at Infinity:Horizontal Asymptotes
Find the horizontal asymptote for the graph of f(x) =
Continuity
• Intuitively, a function is said to be continuousif we can draw a graph of the function with one continuous line.
I. e. without removing our pencil from the graph paper.
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