 Solution

ID: past-exam-of-the-mathematical-methods-ii-course-of-the-university-of-oxford/2018/3/solution

Rationalize the difference of square roots before taking the limit. For positive

x

, divide the resulting numerator and denominator by

x^{2}

\frac{8 x ^{2} + 2 x}{( x ^{3} + 10 x ^{2} + x ^{3} + 2 x ^{2} - 2 x ) 6 x + 3} = \frac{8 + 2/ x}{( 1 + 10/ x + 1 + 2/ x - 2/ x ^{2} ) 6 + 3/ x} .

Each factor now has a finite nonzero denominator limit, giving

\frac{4}{6} = \frac{2 6}{3} .

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