given the following rational, find the horizontal asympotes
the vertical asymptotes if any and the oblique asymptotes if any
f(x)=x^2-x-2 over 2x^2-x-10
f = (x-2)(x+1)/(2x-5)(x+2)
That should make it easy to gather the needed info.
Given the following rational function, find
a. the horizontal asymptote(s),
b. the vertical asymptote(s), if any, and
c. the oblique asymptote(s), if any.
f(x)=(x^2-x-2)/(2x^2-x-10)
To find the horizontal asymptotes, vertical asymptotes, and oblique asymptotes for the given rational function f(x) = (x^2 - x - 2) / (2x^2 - x - 10), we can use the properties of rational functions.
1. Horizontal Asymptotes:
To find the horizontal asymptotes, we need to compare the degrees of the numerator and the denominator.
- If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is at y = 0 (the x-axis).
- If the degree of the numerator is equal to the degree of the denominator, divide the leading coefficients of the numerator and the denominator to obtain the horizontal asymptote.
- If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.
In our case, the degree of the numerator (x^2 - x - 2) is 2, and the degree of the denominator (2x^2 - x - 10) is also 2. So, we need to divide the leading coefficients of the numerator and the denominator.
The leading coefficient of the numerator is 1, and the leading coefficient of the denominator is 2. So, the horizontal asymptote is y = 1/2.
2. Vertical Asymptotes:
To find the vertical asymptotes, we need to determine the values of x when the denominator is zero, but the numerator is not zero. These values of x will give us the vertical asymptotes.
In our case, we need to solve the equation 2x^2 - x - 10 = 0 to find the values of x that make the denominator zero.
Factoring the quadratic equation or using the quadratic formula, we find that the roots of the equation are x = -2 and x = 5.
So, the vertical asymptotes are x = -2 and x = 5.
3. Oblique Asymptotes:
Oblique (or slant) asymptotes occur when the degree of the numerator is exactly one greater than the degree of the denominator.
In our case, the degree of the numerator (2) is one greater than the degree of the denominator (1). So, we can have an oblique asymptote.
To find the equation of the oblique asymptote, we perform polynomial long division by dividing the numerator by the denominator:
(x^2 - x - 2) ÷ (2x^2 - x - 10)
The result of the division is (1/2) + (-2x + 3) / (2x^2 - x - 10).
As x approaches positive or negative infinity, the term involving x in the division result becomes negligible compared to (1/2). Therefore, the oblique asymptote is y = (1/2).
In summary:
- The horizontal asymptote is y = 1/2.
- The vertical asymptotes are x = -2 and x = 5.
- The oblique asymptote is y = (1/2).