Part BCSIR NET June 2023limits-of-perturbed-arguments

Limits of perturbed arguments

Which of the following assertions is correct?

A. $l im s u p_{n} exp (cos ((nπ + (- 1)^{n} \cdot 2 e) / (2 n))) > 1$ .
B. $l i m_{n} exp (l o g_{e} ((n π^{2} + (- 1)^{n} e^{2}) / (7 n)))$ does not exist.
C. $l imin f_{n} exp (sin ((nπ + (- 1)^{n} \cdot 2 e) / (2 n))) < π$ .✓
D. $l i m_{n} exp (tan ((n π^{2} + (- 1)^{n} e^{2}) / (7 n)))$ does not exist.

Solution

All the inner arguments converge: $(nπ + (- 1)^{n} 2 e) / (2 n) \to π /2$ and $(n π^{2} + (- 1)^{n} e^{2}) / (7 n) \to π^{2} /7$ . So the limits in (2) and (4) exist; in (1) the limit is $exp (cos π /2) = 1$ , not > 1; in (3) the limit is $exp (sin π /2) = e < π$ . Official key: 3.

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

Limits of perturbed arguments

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