ASYMPTOTES

Horizontal Vertical

Slantand Holes

Dr.Osama A Rashwan

Definition of an asymptote الخطوط تعريفالتقاربية

An asymptote is a straight line which acts as a boundary for the graph of a function.بالخط يسمى للدالة الممثل للمنحنى حدا يمثل الذي الخط

له التفاربي When a function has an asymptote (and not all

functions have them) the function gets closer and closer to the asymptote as the input value to the function approaches either a specific value a or positive or negative infinity.

خطوط لها التي والدالة تقاربية خطوط لها الدوال كل ليسسالب أو موجب عند أو معينه قيم عند منها تقترب تقاربية

ماالنهاية The functions most likely to have asymptotes are

rational functions كسرية دوال تكون ما غالبا تقاربية خطوط لها التي الدوال

Vertical Asymptotes التقاربية الخطوطالرأسيةVertical asymptotes occur when thefollowing condition is met: The denominator of the simplified rational function is equal to 0. Remember, the simplified rational function has cancelled any factors common to both the numerator and denominator.

العوامل نحذف الكسرية للدالة الرأسية التقاربية الخطوط تحددبالصفر المقام بمساواة ثم أوال ومقامها يسطها بين المشتركة

من عامل كل بوضع الرأسية التقاربية الخطوط معادالت على نحصلصفر يساوي المقام عوامل

Finding Vertical AsymptotesExample 1 1مثال

Given the function

The first step is to cancel any factors common to both numerator and denominator. In this case there are none.

The second step is to see where the denominator of the simplified function equals 0.

x

xxf

22

52

1

01

012

022

x

x

x

x

Finding Vertical Asymptotes Example 1 Con’t. تابع1مثال

The vertical line x = -1 is the only vertical asymptote for the function. As the input value x to this function gets closer and closer to -1 the function itself looks and acts more and more like the vertical line

x = -1.

Graph of Example 1

The vertical dotted line at x = –1 is the vertical asymptote.

Finding Vertical AsymptotesExample 2 2مثال

If

First simplify the

function. Factor

both numerator

and denominator

and cancel any

common factors.

9

121022

2

x

xxxf

3

42

33

423

9

121023

2

x

x

xx

xx

x

xx

Finding Vertical Asymptotes Example 2 Con’t. مثال 2تابع The asymptote(s) occur where the

simplified denominator equals 0.

The vertical line x=3 is the only vertical asymptote for this function.

As the input value x to this function gets closer and closer to 3 the function itself looks more and more like the vertical line x=3.

3

03

x

x

Graph of Example 2

The vertical dotted line at x = 3 is the vertical asymptote

Finding Vertical AsymptotesExample 3 3مثال

If

Factor both the numerator and denominator and cancel any common factors.In this case there are no common factors to cancel.

6

52

xx

xxg

32

5

6

52

xx

x

xx

x

Finding Vertical AsymptotesExample 3 Con’t. مثال 3تابع

The denominator equals zero whenever either

or

This function has two vertical asymptotes, one at x = -2 and the other at x = 3

2

02

x

x

3

03

x

x

Graph of Example 3

The two vertical dotted lines at x = -2 and x = 3 are the vertical asymptotes

Horizontal Asymptotes

Horizontal asymptotes occur when either one of the following conditions is met (you should notice that both conditions cannot be true for the same function).

The degree of the numerator is less than the degree of the denominator. In this case the asymptote is the horizontal line y = 0.

The degree of the numerator is equal to the degree of the denominator. In

this case the asymptote is the horizontal line y = a/b where a is the leading coefficient in the numerator and b is the leading coefficient in the denominator.

When the degree of the numerator is greater than the degree of the denominator there is no horizontal asymptote

الخط معادلة وتكون الالنهاية عند الدالة نهاية قيمة من األفقية التقاربية الخطوط تحددهي األفقي التقاربي

Y= الالنهاية عند الدالة نهاية قيمةدرجة y=0وعليه من أقل البسط درجة تكون عندما األفقي التقاربي الخط معادلة هي

المقام البسط درجة من أقل المقام درجة تكون عندما أفقي تقاربي خط يوجد وال

Finding Horizontal AsymptotesExample 4 4مثال

If

then there is a horizontal asymptote at the line y=0 because the degree of the numerator (2) is less than the degree of the denominator (3). This means that as x gets larger and larger in both the positive and negative directions (x → ∞ and x → -∞)

the function itself looks more and more like the horizontal line y = 0

27

533

2

x

xxxf

Graph of Example 4

The horizontal line y = 0 is the horizontal asymptote.

Finding Horizontal Asymptotes Example 5

If

then because the degree of the numerator (2) is equal to the degree of the denominator (2)

there is a horizontal asymptote at the line y=6/5.

Note, 6 is the leading coefficient of the numerator and 5 is the leading coefficient of the denominator. As x→∞ and as x→-∞ g(x) looks

more and more like the line y=6/5

975

5362

2

xx

xxxg

Graph of Example 5

The horizontal dotted line at y = 6/5 is the

horizontal asymptote.

Finding Horizontal Asymptotes Example 6

If

There are no horizontal asymptotes because the degree of the numerator is greater than the degree of the denominator.

1

9522

3

x

xxxf

Graph of Example 6

Slant Asymptotes

Slant asymptotes occur when the degree of the numerator is exactly one bigger than the degree of the denominator. In this case a slanted line (not horizontal and not vertical) is the function’s asymptote.

To find the equation of the asymptote we need to use long division – dividing the numerator by the denominator.

Finding a Slant Asymptote Example 7

If

There will be a slant asymptote because the degree of the numerator (3) is one bigger than the degree of the denominator (2).

Using long division, divide the numerator by the denominator.

1

9522

23

xx

xxxxf

Finding a Slant AsymptoteExample 7 Con’t.

39521 232

xxxxxx

xxx 23

943 2 xx

333 2 xx

127 x

Finding a Slant AsymptoteExample 7 Con’t.

We can ignore the remainder

The answer we are looking for is the quotient

and the equation of the slant asymptote is

127 x

3x3xy

Graph of Example 7

The slanted line y = x + 3 is the slant asymptote

Holes

Holes occur in the graph of a rational function whenever the numerator and denominator have common factors. The holes occur at the x value(s) that make the common factors equal to 0.

The hole is known as a removable singularity or a removable discontinuity.

When you graph the function on your calculator you won’t be able to see the hole but the function is still discontinuous (has a break or jump).

Finding a HoleExample 8

Remember the

function

We were able to cancel the (x + 3) in the numerator and denominator before finding the vertical asymptote.

Because (x + 3) is a common factor there will be a hole at the point where

33

423

9

121063

2

xx

xx

x

xxxf

3

03

x

x

Graph of Example 8

Notice there is a hole in the graph at the point where x = -3. You would not be able to see this hole if you graphed the curve on your calculator (but it’s there just the same.)

Finding a HoleExample 9

If Factor both numerator and denominator to see if

there are any common factors.

Because there is a common factor of x - 2 there will be a hole at x = 2. This means the function is undefined at x = 2. For every other x value the function looks like

22

422

4

8 2

2

3

xx

xxx

x

xxf

4

82

3

x

xxf

2

422

x

xx

Graph of Example 9

There is a hole in the curve at the point where x = 2. This curve also has a vertical asymptote at x = -2 and a slant asymptote y = x.

Problems

Find the vertical asymptotes, horizontal asymptotes, slant asymptotes and holes for each of the following functions. (Click mouse to see answers.)

2

2

2 15

7 10

x xf x

x x

Vertical: x = -2Horizontal : y = 1Slant: noneHole: at x = - 5

22 5 7

3

x xg x

x

Vertical: x = 3Horizontal : noneSlant: y = 2x +11Hole: none

Just to refresh your memory, consider the following limits.

?

0

4

4)2(

22

4

2lim

222

x

xx

Good job if you saw this as “limit does not exist” indicating a vertical asymptote at x = -2.

?0

0

4)2(

22

4

2lim

222

x

xx

This limit is indeterminate. With some algebraic manipulation, the zero factors could cancel and reveal a real number as a limit. In this case, factoring leads to……

4

1

2

1lim

)2)(2(

2lim

4

2lim

2

222

x

xx

x

x

x

x

xx

The limit exists as x approaches 2 even though the function does not exist. In the first case, zero in the denominator led to a vertical asymptote; in the second case the zeros cancelled out and the limit reveals a hole in the graph at (2, ¼).

x

y

4

2)(

2

x

xxf