What is the vertical and horizontal asymptotes of
..
3x
_________
x^2-3x-10
vertical asymptotes occur when the denominator is zero, so
x^2 - 3x - 10 = 0
(x-5)(x+2) = 0
x = 5 or x = -2 are vertical asymtotes
let x ---> ∞
so 3x/(x^2 - 3x-10) ---> 0
the horizontal asymptote is y = 0 , which is the x-axis
To find the vertical asymptotes of a rational function, you need to determine the values of x that make the denominator equal to zero.
In this case, the denominator is x^2 - 3x - 10. We can factor this expression to find its roots:
x^2 - 3x - 10 = (x - 5)(x + 2)
Setting each factor equal to zero and solving for x, we find:
x - 5 = 0 => x = 5
x + 2 = 0 => x = -2
Therefore, the vertical asymptotes occur when x = 5 and x = -2.
To find the horizontal asymptote, we look at the degrees of the numerator and denominator:
The degree of the numerator is 1 (since the highest power of x is x^1 = x).
The degree of the denominator is 2 (since the highest power of x is x^2).
In this case, the degree of the numerator is less than the degree of the denominator. Therefore, the horizontal asymptote is at y = 0.
So, the vertical asymptotes of the given rational function are x = 5 and x = -2, and the horizontal asymptote is y = 0.
To find the vertical asymptote of a rational function, we need to determine the values that make the denominator equal to zero.
In this case, the denominator x^2 - 3x - 10 factors as (x - 5)(x + 2). So, the vertical asymptotes occur when the denominator equals zero, which gives us x - 5 = 0 or x + 2 = 0. Solving these equations, we find x = 5 and x = -2. Therefore, the vertical asymptotes are x = 5 and x = -2.
To find the horizontal asymptote of a rational function, we consider the degrees of the numerator and denominator. In our case, the degree of the numerator is 1 (since the highest power of x is 1) and the degree of the denominator is 2 (since the highest power of x is 2). When the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y = 0.
So, the horizontal asymptote of the given rational function is y = 0.