Limits

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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