Question

The vertical and horizontal asymptotes to the curve Y = $\dfrac{{\log x}}{x}$ ($\forall $x > 0) are respectively.
$\left( A \right)$ x = 1, y = 0
$\left( B \right)$ x = 0, y = 0
$\left( C \right)$ x = 0, y = 1
$\left( D \right)$ x = 1, y = 1

Answer 1

Hint – In This question use the concept that for vertical asymptotes equate the denominator of the curve to zero and solve for x and for the horizontal asymptotes If the degree of the numerator is less than the degree of the denominator than the horizontal asymptotes is the x-axis (i.e. y = 0) so use these concepts to reach the solution of the question.

Complete step-by-step answer:
Given curve:
Y = ($\dfrac{{\log x}}{x}$)
Vertical asymptotes:
Vertical asymptotes are the vertical lines corresponding to the zeroes of the denominator of the rational number, to calculate the vertical asymptotes simply put the denominator of the rational number to zero, and solve for x, this is the required vertical asymptotes, if the denominator of the rational number is a quadratic, cubic types equations or any other polynomial then simplify put the whole equation to zero and simplify used any method for x, this is the required vertical asymptotes.
So in the given curve, Y = ($\dfrac{{\log x}}{x}$)
The denominator is simply x, so equate this to zero,
Therefore, x = 0 so this is the required solution for the vertical asymptotes.
So, x = 0 is the vertical asymptotes.
Horizontal asymptotes:
As we know that the horizontal asymptotes are the horizontal lines that the graph of the rational number approaches.
There are two ways to calculate the horizontal asymptotes:
$\left( 1 \right)$ If the degree of the numerator is less than the degree of the denominator than the horizontal asymptotes is the x-axis (i.e. y = 0).
$\left( 2 \right)$ If the degree of the numerator is greater than the degree of the denominator than there is no horizontal asymptote.
So in the given graph the denominator is x so the degree of the denominator is 1.
And the numerator is log x, so the degree of the log x we all know i.e. zero (0).
So as we see that the degree of the numerator is less than the degree of the denominator than the horizontal asymptotes is the x-axis (i.e. y = 0).
So the horizontal asymptotes is (y = 0).
So this is the required answer.
Hence option (B) is the correct answer.

Note – Whenever we face such types of questions the key concept we have to remember is that vertical asymptotes are the solution in terms of x and the horizontal asymptotes are the solution in terms of y, so use the definition of vertical and horizontal asymptotes which is stated above and solve for x and y, we will get the required vertical and horizontal asymptotes.