Question

STATEMENT-1 : ${lim}_{x\to \infty }(\frac{1^2}{x^3}+\frac{2^2}{x^3}+\frac{3^2}{x^3}+.......+\frac{x^2}{x^3})=\underset{x\to \infty }{lim}\frac{1^2}{x^3}+\underset{x\to \infty }{lim}\frac{2^2}{x^3}+.....+\underset{x\to \infty }{lim}\frac{x^2}{x^3}=0$
STATEMENT-2 : ${lim}_{x\to a}\left(f_1\right(x)+f_2(x)+.....+f_n(x\left)\right)={lim}_{x\to a}{f}_1\left(x\right)+{lim}_{x\to a}{f}_2\left(x\right)+....+{lim}_{x\to a}{f}_n\left(x\right)$, where $n\in N.$

• A. STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is a correct explanation for STATEMENT-1
• B. STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is NOT a correct explanation for STATEMENT-1
• C. STATEMENT-1 is True, STATEMENT-2 is False
• D. STATEMENT-1 is False, STATEMENT-2 is True

Answer 1

The correct option is D STATEMENT-1 is False, STATEMENT-2 is True

Reason is true (from theory )

Assertion:

$\underset{x\to \infty }{lim}(\frac{1^2}{x^3}+\frac{2^2}{x^3}+\frac{3^2}{x^3}+....+\frac{x^2}{x^3})$

$=\underset{x\to \infty }{lim}\left(\sum _1^x\frac{r^2}{x^3}\right)$

$=\underset{x\to \infty }{lim}\frac{x(x+1)(2x+1)}{6x^3}$

$=\underset{x\to \infty }{lim}\frac{(1+\frac{1}{x})(2+\frac{1}{x})}{6}=\frac{1}{3}$