How do you find the asymptotes for #f(x)=x/(4x^2+7x-2)#?

Question

Sadie Ackerman · Accepted Answer

vertical asymptotes , x = -2 , #x = 1/4#
horizontal asymptote y = 0

Vertical asymptotes occur when the denominator of a rational function tends to zero. To find the equation/s let the denominator equal zero. solve : # 4x^2+7x-2 = 0 → (4x-1)(x+2) = 0 # # rArr x = - 2 , x = 1/4" are the asymptotes " # Horizontal asymptotes occur as # lim_(x→±∞) f(x) → 0 # If the degree of the numerator is less than the degree of the denominator, as in this question, then the equation of the asymptote is always y = 0. Here is the graph of f(x) graph{x/(4x^2+7x-2) [-10, 10, -5, 5]}

Serenity Johnson · Answer

undefined To find the asymptotes of the function $ f(x) = \frac{x}{4x^2 + 7x - 2} $, you need to check for vertical, horizontal, and slant asymptotes.

1. Vertical Asymptotes:
   Vertical asymptotes occur where the denominator of the rational function is zero but the numerator is not. Find the values of $ x $ that make the denominator $ 4x^2 + 7x - 2 = 0 $ by solving the quadratic equation.

2. Horizontal Asymptotes:
   Horizontal asymptotes can be determined by examining the degrees of the numerator and denominator. If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is $ y = 0 $. If the degrees are equal, the horizontal asymptote is the ratio of the leading coefficients. If the degree of the numerator is greater, there is no horizontal asymptote.

3. Slant Asymptotes:
   Slant (or oblique) asymptotes occur when the degree of the numerator is one greater than the degree of the denominator. To find the equation of the slant asymptote, perform polynomial long division or synthetic division to divide the numerator by the denominator. The quotient obtained represents the equation of the slant asymptote.

After determining the asymptotes using these methods, you'll have a complete understanding of the behavior of the function as $ x $ approaches positive or negative infinity.