Vertical, Horizontal, and Slant Asymptotes

Asymptotes of Rational Functions A rational function has the form f(x)=\frac{P(x)}{Q(x)}, where P(x) and Q(x) are polynomials and Q(x)\neq 0. 1. Discontinuities Discontinuities occur where the denominator is zero, i.e., where Q(x)=0. Removable discontinuity (hole): If a factor cancels between numerator and denominator, the discontinuity is just a hole in the graph. Essential discontinuity: If the factor does not cancel, then the function “blows up” near that value → this gives a vertical asymptote. 2. Vertical Asymptotes Occur at values of x where Q(x)=0 after simplification. The function approaches +infinity or -infinity near these values. 3. Horizontal Asymptotes These describe the behavior of the function as x \to \pm \infty. Let: Degree of P(x) = n Degree of Q(x) = m Cases: 1. If n is less than m: y = 0 (The graph approaches the x-axis) 2. If n = m: y = leading coefficient of P/ leading coefficient of Q 3. If n is greater than m: No horizontal asymptote. 4. Slant (Oblique) Asymptotes Occur when the degree of the numerator is exactly one more than the denominator: n = m + 1 To find it: Perform polynomial long division and the quotient is the slant asymptote. Big Picture Summary Vertical asymptotes: where the function is undefined and doesn’t cancel Horizontal asymptotes: based on comparing degrees of numerator and denominator Slant asymptotes: when numerator degree is one higher → use long division. You can summarize it for students like this: First simplify. Then: Denominator = 0 gives vertical asymptote (unless it cancels) Compare degrees gives horizontal asymptote If top is exactly one degree higher so divide and find the quotient as slant asymptote