 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2003/paper-1/2/a/i/solution

Codex  0 Created 2026-10-18 Updated 2026-10-20

Write

R (x) = (x^{2} + 3) (2 x^{2} - 1) / (3 x^{4} - x^{3} - 1)

. This rational function is continuous at both finite points because its denominator is nonzero there. Substitution gives the first two limits; at infinity divide numerator and denominator by

x^{4}

and use their leading coefficients. Therefore

x \to 0 lim R (x) = 3, x \to 1 lim R (x) = 4, x \to \infty lim R (x) = \frac{2}{3} .

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