Horizontal Asymptotes and End Behavior

Calculus 1 · Axiom Academy

LESSON Horizontal Asymptotes & End Behavior Where a rational function flattens out as — decided by a contest between the leading terms. The statement asks a simple question: as x marches off to the right forever, what height does f(x) settle toward? If it homes in on a single value L , the line y = L is a horizontal asymptote — the graph hugs it more and more closely out at the ends. A horizontal asymptote describes behavior at the ends Write a rational function as , with and . Far from the origin, each polynomial behaves like its leading term , so the whole question reduces to comparing n and m — three cases, nothing more. Here a_n and b_m are the leading coefficients of P and Q . They matter only in the tie ( n = m ); in the other two cases the degree gap alone settles it. Take . The numerator's leading term is 3x^2 (degree 2 ); the denominator's is 4x^3 (degree 3 ). Since x^3 grows far faster than x^2 , the denominator outpaces the numerator and the fraction is squeezed toward 0 . 4. A Tie: Ratio of Leading Coefficients Now . Both top and bottom have degree 3 , with leading terms 6x^3 and 2x^3 . They grow at the same rate, so neither runs away — instead the curve settles at the ratio of the leading coefficients . 5. Numerator Wins: No Horizontal Asymptote Finally . The numerator has degree 4 ; the denominator only degree 2 . The top's x^4 dominates, so the fraction grows without bound — there is no horizontal line the graph approaches. HA at (ratio of leading coefficients)

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