How do you find vertical, horizontal and oblique asymptotes for #(x^2-4)/(x)#?

Question

Christian Antunez · Accepted Answer

vertical asymptote x = 0
oblique asymptote y = x

When the denominator of a rational function tends to zero, vertical asymptotes occur. Let the denominator equal zero to find the equation. Consequently, the asymptote is x = 0. Horizontal asymptotes occur as #lim_(x→±∞) f(x) → 0 There will be an oblique asymptote rather than a horizontal asymptote when the degree of the numerator is higher than the degree of the denominator. now # (x^2-4)/x = x^2/x - 4/x = x - 4/x # As x→±∞ , # 4/x → 0 " and " y → x# # rArr y = x " is an oblique asymptote " # graph{(x^2-4)/x [-10, 10, -5, 5]}

Sadie Ackerman · Answer

undefined To find the vertical asymptotes, set the denominator equal to zero and solve for x. In this case, solve x = 0.

To find the horizontal asymptotes, compare the degrees of the numerator and denominator. Since the degree of the numerator is 2 and the degree of the denominator is 1, there is no horizontal asymptote.

To find the oblique asymptotes, perform polynomial long division or synthetic division of the numerator by the denominator. In this case, divide x^2 - 4 by x. This yields x - 4 as the quotient. Therefore, the oblique asymptote is y = x - 4.