Question

Consider the following statements:
$S_1$:$\underset{x\to 0^{-}}{lim}\frac{\left[x\right]}{x}$ is an indeterminate from (where [.] denotes greatest integer function).
$S_2$: $\underset{x\to \infty }{lim}\frac{\sin \left(3^x\right)}{3^x}=0$
$S_3$: $\underset{x\to \infty }{lim}\sqrt{\frac{x-\sin x}{x+{\cos }^{2}x}}$ does not exist.
$S_4$: $\underset{n\to \infty }{lim}\frac{(n+2)!+(n+1)!}{(n+3)!}(n\in N)=0$
Which of the statements $S_1,S_2,S_3,S_4$ are true or false:

• A. $FTFT$
• B. $FTTT$
• C. $FTFF$
• D. $TTFT$

Answer 1

The correct option is D $TTFT$

$\to$ For $x\to 0,\left[x\right]=-1,\Rightarrow {lim}_{x\to 0}\frac{-1}{x}$ is

indeterminat

only option with S, true is

(D) $\Rightarrow \left(D\right)$