Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 2 / 32E / a

Table of contents
- i a
  - Solution i
- ii a
  - Solution ii
- iii a
  - Solution iii

i

Solution

This is an asymptotic sequence. Since

lo g (cos u) \sim - \frac{u ^{2}}{2} (u \to 0),

one has

\frac{ϕ _{n + 1} ( x )}{ϕ _{n} ( x )} \sim x^{2} ⟶ 0.

ii

Solution

This is not an asymptotic sequence, because

\frac{ψ _{n + 1} ( x )}{ψ _{n} ( x )} = (\frac{n + 1}{n})^{1/ x} ⟶ 1

x \to \infty

, rather than zero.

iii

Solution

This is not an asymptotic sequence. The functions

sin (x^{n})

oscillate and have arbitrarily large zeros; the ratio

sin (x^{n + 1}) / sin (x^{n})

has no limit, let alone limit zero.

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