Introduction
An asymptote is a line that approaches closer to a given curve as one or both of xory coordinates tend to infinity but never intersects or crosses the curve.
There are two types of asymptotes viz. Rectangular asymptotes and Oblique asymptotes.
A rectangular asymptote is parallel to x-axis or toy-axis. If an asymptote is parallel to x-axis then it is called horizontal asymptote and if the asymptote parallel to y-axis then it is called vertical asymptote.
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6? Horizontal Asymptote
Vertical Asymptote
If lim
x→a−
f(x) =±∞or lim
x→a+
f(x) =±∞then the line x=ais said to be a vertical asymptote of the graph ofy=f(x).
If lim
x→∞f(x) = b or lim
x→−∞f(x) = b then the line y = b is said to be a horizontal asymptote of the graph ofy=f(x).
It should be noted that a graph can have an infinite number of vertical asymptotes, but it can only have at most two horizontal asymptotes.
Horizontal asymptotes describe the left and right-hand behavior of the graph.
A graph will (almost) never touch a vertical asymptote; however, a graph may cross a horizontal asymptote.
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6?
Oblique or Slant Asymptote
An oblique asymptote or slant asymptote is an asymptote that is neither parallel to x-axis nor to y-axis. A oblique asymptote, just like a horizontal asymptote, guides the graph of a function only when xis close to ±∞but it is a slanted line, i.e. neither vertical nor horizontal. A rational function has a slant asymptote if the degree of a numerator polynomial is one more than the degree of the denominator polynomial.
Method of finding asymptote Letf(x) be a rational function.
Then
f(x) = anxn+an−1xn−1+· · ·+a1x+a0
bmxm+bm−1xm−1+· · ·+b1x+b0
The graph ofy=f(x) has vertical asymptotes at those values ofxfor which the denominator is equal to zero.
The graph of y=f(x) will have at most one horizontal asymptote.
i. Ifm > n(that is, the degree of the denominator is larger than the degree of the numerator), then the graph ofy=f(x) will have a horizontal asymptote at y = 0 (i.e., the x-axis).
ii. Ifm=n(that is, the degrees of the numerator and denominator are the same), then the graph ofy=f(x) will have a horizontal asymptote aty=ban
m. iii. Ifm < n(that is, the degree of the numerator is larger than the degree of the denominator), then the graph ofy=f(x) will have no horizontal asymptote.
Example 1 Find the vertical and horizontal asymptotes of the graph of f(x) = 2x+ 1
x2−1. Solution: Putting Denominator= 0 , we get
x2−1 = 0 or x=±1
Therefore, the vertical asymptotes are x= 1 and x=−1.
Here, the degree of the denominator is larger than the degree of the numer- ator. So the given curvey=f(x) has the horizontal asymptotey= 0.
Example 2 Find the vertical and horizontal asymptotes of the graph of f(x) = x−7
x2−4x−21.
Solution: Again, we start with the zeros of the denominator.
x2−4x−21 = 0⇒(x−7)(x+ 3) = 0 or x−7 = 0 and x+ 3 = 0 x= 7 and x=−3
Now, the numerator becomes zero forx= 7
Therefore, here we have one vertical asymptote: x=−3.
Note: Thex= 7 gives us a hole in the graph.
A recipe for finding a slant asymptote of a rational function:
Divide the numerator by the denominator. Use long division of polynomials or, in case of denominator being of the form: (x−c), you can use synthetic division. The equation of the asymptote isy=mx+b which is the quotient of the polynomial division
Example 3 Find slant of the curve
f(x) = 2x2+x−5 x+ 1 . Solution: The given equation can be expressed as
f(x) = 2x−1 + −4 x+ 1 The required asymptote isy= 2x−1.
Method of finding asymptote of polynomial Equation
Asymptote parallel tox-axis is obtained by equating the coefficient of highest power ofxin the equation of the curve to zero.
Asymptote parallel toy-axis is obtained by equating the coefficient of highest power ofy in the equation of the curve to zero.
Example 4 Find the vertical and horizontal asymptotes of the curve
a2x2+b2y2=x2y2.
Solution: The given equation of the curve can be written as x2(y2−a2)−b2y2= 0
Here highest available power ofxis 2. Equating coefficient ofx2to zero, we get y2−a2= 0⇒y=±a
Thereforey=aandy=−aare two horizontal asymptotes.
Again, equating coefficient ofy2to zero, we get x2−b2= 0⇒x=±b Thereforex=b andx=−bare two vertical asymptotes.
To find the oblique asymptote proceed as follows:
Let the asymptote bey=mx+c.
Step 1: Express the equation of the given curve in the form ϕn(x, y) +ϕn−1(x, y) +· · ·+ϕ2(x, y) +ϕ1(x, y) +c= 0 whereϕn(x, y) is highest degree (i.e. nth) term of the curve.
Step 2: Put x= 1, y=minϕn(x, y),ϕn−1(x, y),. . ., ϕ(x, y).
Step 3: Find all the real roots ofϕn(m) = 0.
Step 4: Ifmis a simple root, then corresponding value ofc is given by cϕ′n(m) +ϕn−1(m) = 0, ϕ′n(m)̸= 0
If ϕ′n(m) = 0 then there is no asymptote to the curve corresponding to this value ofm.
Step 5: Ifmis a multiple root of order two, then the two values ofcare given by
c2
2!ϕ′′n(m) + c
1!ϕ′n−1(m) +ϕn−2= 0, ϕ′′n(m)̸= 0
Note: Ifmis a multiple root of orderrthen thervalues ofcare given by cr
r!ϕn(r)
(m) + cr−1
(r−1)!ϕn−1(r−1)
(m) +· · ·+ c
1!ϕ′n−r−1(m) +ϕn−r(m) = 0.
Example 5 Find the asymptotes of
x3+x2y−xy2−y3+x2−y2−2 = 0
Solution: Here the coefficients of highest available power ofxandyin the given curve are constants. Therefore there are no horizontal or vertical asymptote.
To fin the oblique asymptotes, putx= 1 andy=minϕ3(x, y) =x3+x2y− xy2−y3, the highest degree term.
ϕ3= 0⇒1 +m−m2−m3= 0 or m=−1,−1,1 ϕ′3(m) = 1−2m−3m2 Form= 1,
c=−ϕ2(1) ϕ′3(1) = 0 Thereforey=xis an asymptote.
Since−1 is a double root ofϕ3(m) = 0, the values ofccan be obtained from c2
2!ϕ′′3(−1) + c
1!ϕ′2(−1) +ϕ1(−1) = 0 or 2c2+ 2c= 0
or c= 0,−1
Therefore, the required asymptotes arey=x,y=−x,y=−x−1.
Exercise
1. Find the vertical and horizontal asymptotes of the graph of f(x) = x
x2+ 1. 2. Find the asymptotes of the following curves
(i)y3+x2y+ 2xy2−y+ 1 = 0
(ii)x4−5x2y2+ 4y4+x2−2y2+ 2x+y+ 7 = 0
(iii)x3−x2y−xy2+y3+ 2x2−4y2+ 2xy+x+y+ 1 = 0 (iv)ax2+ 2hxy+by2+ 2gx+ 2f y+c= 0
