Skip to content
Part BCSIR NET June 2023limits-of-perturbed-arguments

Limits of perturbed arguments

Which of the following assertions is correct?

  1. A.limsupnexp⁡(cos⁡((nπ+(−1)n⋅2e)/(2n)))>1limsup_{n} \exp(\cos((n\pi + (-1)^{n}\cdot2e)/(2n))) > 1.
  2. B.limnexp⁡(loge((nπ2+(−1)ne2)/(7n)))lim_{n} \exp(log_{e}((n\pi^{2} + (-1)^{n}e^{2})/(7n))) does not exist.
  3. C.liminfnexp⁡(sin⁡((nπ+(−1)n⋅2e)/(2n)))<πliminf_{n} \exp(\sin((n\pi + (-1)^{n}\cdot2e)/(2n))) < \pi.✓ Correct
  4. D.limnexp⁡(tan⁡((nπ2+(−1)ne2)/(7n)))lim_{n} \exp(\tan((n\pi^{2} + (-1)^{n}e^{2})/(7n))) does not exist.

You have the answer. Trap Analysis is why the other three were written.

Not a worked solution repeated four times — the specific reasoning error each wrong option was built to reward, and why this one tests limit assumed to exist.

See pricing

50 are analysed free — try those first.

The trap it tests

Limit assumed to exist

Reasoning with a limit before establishing there is one. limsup is not lim.

Drill statements like this

Related counterexample: limsup(aₙ + bₙ) = limsup aₙ + limsup bₙ

More on this topic

The chapter behind this: limsup and liminf — three equivalent definitions and the algebra — free to read

From The Real Line › limsup, liminf and subsequential limits

Last revised . Found a mistake? Tell us — corrections are the fastest thing we act on.

ShareWhatsAppTelegram