Comments by "MC116" (@angelmendez-rivera351) on "Calculus Limits at Infinity Rational Functions Four Easy Examples" video.

  1. Technically, a limit to infinity is a limit to 0 of the limiting function composed from the right with the reciprocal function. Symbolically, we define h : R\{0} —> R\{0}, h(x) == 1/x, and we define lim f (x —> +♾) := lim f°h (x > 0, x —> 0), and lim f (x —> –♾) := lim f°h (x < 0, x —> 0). This actually helps clarify what is going on. If you have polynomials P and Q, then lim P(x)/Q(x) (x —> +♾) := lim P(1/x)/Q(1/x) (x > 0, x —> 0). Now, let P' denote the polynomial that has coefficients of P in the reversed order. So the nth coefficient of P' is the [deg(P) – n]-th coefficient of P. Let Q' be defined similarly. Then lim P(1/x)/Q(1/x) (x > 0, x —> 0) = lim [P'(x)/x^deg(P)]/[Q'(x)/x^deg(Q)] (x > 0, x —> 0) = lim P'(x)/Q'(x)·x^[deg(Q) – deg(P)] (x > 0, x —> 0). With this, whether the limit exists, and what its value is, depends on the value of deg(Q) – deg(P) and on the value of the lowest order nonzero coefficients of P' and Q'.
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