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

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- Solution i

Solution

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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