Asymptote Formula
Asymptote Formula
A straight line in geometry known as an Asymptote Formula tends to intersect a curve at infinity when it approaches the curve on the graph. The relationship between the curve and its asymptote is unusual in that they move parallel to one another but never cross anywhere other than infinity.
Furthermore, although they run side by side, they remain apart. There are three different asymptotes: oblique, vertical, and horizontal. As x approaches infinity, the curve approaches a constant value b, where the horizontal Asymptote Formula is found.
When x moves in the direction of infinity as it approaches a constant value c from the right or left, this is where a vertical asymptote is found. This is known as an oblique Asymptote Formula when the curve moves along the y = mx + b line while x also moves in any direction towards infinity.
Horizontal asymptote
As x approaches infinity, the curve approaches a constant value, b, which is where the horizontal asymptotes are found. The horizontal asymptotes of the curve f (x) = (axm +…)/(bxn +..) are as follows:
As x tends to infinity, or limx f(x) = 0, the horizontal asymptote is y = 0 if m n. If m = n, limx f(x) = a/b is the horizontal asymptote as x tends to infinity. There is no horizontal asymptote for the function f(x) if m > n. limx f(x) equals.
Vertical asymptote
When x moves in the direction of infinity as it approaches a constant value c from the right or left, this is where a vertical Asymptote Formula is found. Since a function is undefined when its denominator is zero, the denominator of a function must be equal to zero in order to determine its vertical Asymptote Formula.
Oblique asymptote
An oblique asymptote develops when the curve moves in the direction of the line y = mx + b and x also moves in any direction towards infinity.
Consider the polynomial functions p(x) and q(x) in equation f(x) = p(x)/q(x). Only if the numerator’s degree exceeds the denominator will the provided function have an oblique asymptote. Polynomial division on the above function yields f(x) = a(x) + r(x)/q(x), where a(x) is the quotient and r(x) is the reminder. The provided function’s oblique asymptote is a (x).
Asymptotes of a hyperbola
Two Asymptote Formula of a hyperbola have the equation x2/a2 – y2/b2 = 0.
Solved Example
Find the vertical asymptote of the curve f(x) = 5×2 – 4x + 1/x2 – 5x + 6.
Solution:
Given,
the curve f(x) equation = 5×2 – 4x + 1/x2 – 5x + 6
As we know, a vertical asymptote happens when the curve tends to infinity.
Therefore, zero should be used as the denominator.
⇒ x2 – 5x + 6 = 0
⇒ x2 – 2x – 3x + 6 = 0
⇒ (x – 2) (x – 3) = 0
⇒ x = 2 or x = 3
Hence, the vertical asymptotes are x = 2 and x = 3.
