Question

Assertion :Let $a_n=\underset{ntimes}{\underset{}{2.99\dots 9}}$, $nϵN$. Then, $\left[\underset{n\to \infty }{lim}{a}_n\right]=\underset{n\to \infty }{lim}[a_n]$, $[.]$ denotes the greatest integer function. Reason: $\underset{n\to \infty }{lim}{a}_n=3$

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is incorrect but Reason is correct
• D. Both Assertion and Reason are incorrect

Answer 1

The correct option is C Assertion is incorrect but Reason is correct

$a_n=\mathrm{2.9999...}\left(ntimes\right)$

$\underset{x_n\to \infty }{lim}{a}_n=3$

but as $a_n\to 3$ but still $<3$

$\underset{x_n\to \infty }{lim}\left[a_n\right]=2$

So $\left[\underset{x_n\to \infty }{lim}{a}_n\right]=\left[3\right]=3\ne \underset{x_n\to \infty }{lim}\left[a_n\right]$ which is equal to $2$

Assertion is incorrect, reason is correct.