Which of the following assertions is correct?

Question

Question

This question was previously asked in

CSIR UGC (NET) Mathematical Science: Held On (7 June 2023)

\(\rm\displaystyle\lim _n \sup e^{\cos \left(\frac{n \pi+(-1)^n2e}{2 n}\right)}>1\)
\(\rm\displaystyle\lim _n ~e^{\log _e\left(\frac{n \pi^2+(-1)^n e^2}{7 n}\right)}\) does not exist.
\(\rm\displaystyle\lim _n \inf ~e^{\sin \left(\frac{n \pi+(-1)^n 2 e}{2 n}\right)}<\pi\)
\(\rm\displaystyle\lim _n ~e^{\tan \left(\frac{n \pi^2+(-1)^n e^2}{7 n}\right)}\) does not exist.

Answer 1

Explanation:

= =

So = = e⁰ = 1 1

Option (1) is false

Also = = e < π

Option (3) is correct

Also = =

So = which is finite

Option (2) is false

= which is finite

Option (4) is false

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