Finding All Asymptotes of f(x) = (2x² + 3x - 1)/(x² - 4)

Algebra 2 · Axiom Academy

EXAMPLE Finding All Asymptotes of Systematically finding every vertical, horizontal, and oblique asymptote of a rational function, step by step. Find every vertical, horizontal, and oblique asymptote of , and determine whether its graph has any holes. Nice work! You've completed a full asymptote analysis of a rational function. Here's the process that always works: Factor the denominator first — its zeros are the only candidates for vertical asymptotes. Check for common factors with the numerator before declaring a vertical asymptote — a shared factor cancels into a hole, not an asymptote. Compare the degrees of numerator and denominator: numerator degree less than denominator degree gives y=0 ; equal degrees give y= the ratio of leading coefficients; numerator degree exactly one more than denominator degree gives an oblique asymptote; anything else gives neither. A function never has both a horizontal and an oblique asymptote — the degree comparison always picks exactly one, or neither. Complete answer: has vertical asymptotes at x=-2 and x=2 , a horizontal asymptote at y=2 , no oblique asymptote, and no holes. This same process — factor, check for holes, compare degrees, compute — finds every asymptote of any rational function.

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