Rational functions often approach a finite limit as x becomes very large, yet this behavior is not always immediately obvious from the full algebraic expression.
Consider the function
f(x)=5x3−7x+43x3+2x−1.
(a) Predict the value of x→∞limf(x) without expanding or performing polynomial long division, and provide a verbal justification for your prediction that references specific structural features of the numerator and denominator.
(b) Let P(x) and Q(x) be non-zero polynomials of the same degree n with respective leading coefficients an and bn. Explain, in complete sentences and without carrying out explicit calculations, why
x→∞limQ(x)P(x)=bnan.
Your explanation should reference the relative influence of higher- and lower-degree terms as x grows without bound.
a. Part (a) — Prediction and justification
b. Part (b) — General explanation for polynomials of equal degree