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