Horizontal Asymptotes :
A horizontal asymptote is a horizontal line that is not part of a graph of a function but guides it for x – values “far” to the right and/or “far” to the left. The graph may cross it but eventually, for large enough or small enough values of x (approaching ±∞), the graph would get closer and closer to the asymptote without touching it.
A horizontal asymptote is a special case of a slant asymptote.
Let
deg N(x) = the degree of a numerator
and
deg D(x) = the degree of a denominator
Case 1 :
degree of numerator = degree of denominator
y = leading coefficient of N(x)/leading coefficient of D(x).
Case 2 :
degree of numerator < degree of denominator
y = 0 which is the x – axis.
Case 3 :
degree of numerator > degree of denominator
There is no horizontal asymptote.
Find the horizontal asymptote of the graph of each rational function.
Problem 1 :
y = 2/(x – 6)
Solution :
y = 2/(x – 6)
Degree of numerator = 0
Degree of denominator = 1
degree of numerator < degree of denominator
So, equation of the horizontal asymptote is y = 0 which is the x – axis.
Problem 2 :
y = (x + 2)/(x – 4)
Solution :
y = (x + 2)/(x – 4)
Degree of numerator = 1
Degree of denominator = 1
degree of numerator = degree of denominator
y = leading coefficient of N(x)/leading coefficient of D(x).
So, equation of the horizontal asymptote is y = 1.
Problem 3 :
y = (x + 3)/2(x + 4)
Solution :
Given, y = (x + 3)/2(x + 4)
Degree of numerator = 1
Degree of denominator = 1
degree of numerator = degree of denominator
y = leading coefficient of N(x)/leading coefficient of D(x).
So, equation of the horizontal asymptote is y = 1/2.
Problem 4 :
y = (2x^{2} + 3)/(x^{2} – 6)
Solution :
y = (2x^{2} + 3)/(x^{2} – 6)
Degree of numerator = 2
Degree of denominator = 2
degree of numerator = degree of denominator
y = leading coefficient of N(x)/leading coefficient of D(x).
So, equation of the horizontal asymptote is y = 2.
Problem 5 :
y = (3x - 12)/(x^{2} – 2)
Solution :
Given, y = (3x - 12)/(x^{2} – 2)
Degree of numerator = 0
Degree of denominator = 2
degree of numerator < degree of denominator
So, equation of the horizontal asymptote is y = 0 which is the x – axis.
Problem 6 :
y = (3x^{3 }– 4x + 2)/(2x^{3} + 3)
Solution :
y = (3x^{3 }– 4x + 2)/(2x^{3} + 3)
Degree of numerator = 3
Degree of denominator = 3
degree of numerator = degree of denominator
y = leading coefficient of N(x)/leading coefficient of D(x).
y = 3/2
So, equation of the horizontal asymptote is y = 1.5.
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