3F (Analysis I)

Rationalizing avoids cancellation between the two large terms of the sequence:

a_{n} = \frac{n}{n + n ^{2} - n} = \frac{1}{1 + 1 - 1/ n} .

By continuity of the square root and the reciprocal at the limiting values,

n \to \infty lim a_{n} = \frac{1}{2} .

b

Separate off the integer part of the phase. Set

d_{n} = n^{2} + n - n

. Rationalization gives

d_{n} = \frac{1}{1 + 1/ n + 1} ⟶ \frac{1}{2} .

Since squaring removes the sign in

cos (πn + u) = (- 1)^{n} cos u

, the given sequence satisfies

b_{n} = cos^{2} (π d_{n})

. The continuity of the cosine now gives

n \to \infty lim b_{n} = cos^{2} (π /2) = 0.

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