How do you find all the asymptotes for function #f(x) = (3 - x) / x^2#?

Question

Anna Allen · Accepted Answer

Vertical asymptote: x = 0. When x -> 0, y  -> +infinity Horizontal asymptote y = 0. When x -> -infinity y --> 0 (from + values)
When x  -> +infinity, y --> 0 (from - values) graph{(3 - x)/x^2 [-10, 10, -5, 5]}

Ariana Alvarez · Answer

undefined To find all the asymptotes for the function $ f(x) = \frac{3 - x}{x^2} $, follow these steps:

1. Identify vertical asymptotes by finding values of $ x $ for which the function approaches positive or negative infinity.
2. Determine horizontal asymptotes by analyzing the behavior of the function as $ x $ approaches positive or negative infinity.
3. Find any oblique (slant) asymptotes if they exist.

Let's proceed with these steps:

1. **Vertical Asymptotes**: Set the denominator equal to zero and solve for $ x $ to find vertical asymptotes. In this case, $ x^2 = 0 $ has one root at $ x = 0 $. So, $ x = 0 $ is a vertical asymptote.

2. **Horizontal Asymptotes**: Check the behavior of the function as $ x $ approaches positive and negative infinity. Divide the leading term of the numerator by the leading term of the denominator. In this case, both the numerator and denominator have leading terms of degree 1. Therefore, the horizontal asymptote is given by the ratio of the leading coefficients, which is $ \frac{-1}{1} = -1 $.

3. **Oblique (Slant) Asymptotes**: Check if the degree of the numerator is one greater than the degree of the denominator. If it is, perform polynomial long division to find the oblique asymptote. In this case, the degree of the numerator is less than the degree of the denominator, so there are no oblique asymptotes.

Therefore, the asymptotes for the function $ f(x) = \frac{3 - x}{x^2} $ are:

- Vertical asymptote: $ x = 0 $
- Horizontal asymptote: $ y = -1 $