Question

How to find the asymptotes of \[y = \dfrac{{7x - 5}}{{2 - 5x}}\] ?

Answer 1

Hint: Asymptote is a line that a curve approaches as it moves towards infinity. To find the horizontal asymptote involves comparing the degrees of the polynomials in the numerator and denominator of the function. If both the polynomials have the same degree, divide the coefficients of the largest degree terms. Hence, applying this we get Horizontal Asymptote.

Complete step by step solution:
Let us write the given expression:
 \[y = \dfrac{{7x - 5}}{{2 - 5x}}\] …………………… 1
To find the vertical asymptote equate the denominator terms of equation 1 to zero as:
 \[2 - 5x = 0\]
Now, simplify the terms to get the value of x as:
 \[ \Rightarrow - 5x = - 2\]
 \[ \Rightarrow - x = - \dfrac{2}{5}\]
Hence, we get:
 \[ \Rightarrow x = \dfrac{2}{5}\]
Hence, \[x = \dfrac{2}{5}\] is the equation of the vertical asymptote.
For finding the Horizontal asymptotes, we need to compare the degrees of the numerator and denominator i.e., if the degree of the numerator equals the degree of the denominator then the horizontal asymptote is given by:
y = Lead Coefficient of the Numerator \[ \div \] Lead Coefficient of the Denominator.
Here, in the given expression; both numerator and denominator have the degree of 1, hence from equation 1 we get:
 \[ \Rightarrow y = \dfrac{7}{{ - 5}}\]
 \[ \Rightarrow y = - \dfrac{7}{5}\]
Therefore, the Horizontal Asymptote is \[y = - \dfrac{7}{5}\] .

Note: The method to identify the horizontal asymptote changes based on how the degrees of the polynomial in the function’s numerator and denominator are compared. The point to note is that the distance between the curve and the asymptote tends to be zero as it moves to infinity. Here, in the given expression; both numerator and denominator have the degree of 1, hence we have divided the coefficients of highest degree terms to get the horizontal asymptotes.