Question

Statement $I:\underset{x\to \infty }{\text{lim}}(\frac{1^2}{x^3}+\frac{2^2}{x^3}+\frac{3^2}{x^3}+\cdots +\frac{x^2}{x^3})=\underset{x\to \infty }{\text{lim}}\frac{1^2}{x^3}+\underset{x\to \infty }{\text{lim}}\frac{2^2}{x^3}+\cdots +\underset{x\to \infty }{\text{lim}}\frac{x^2}{x^3}=0$

Statement $II:\underset{x\to a}{\text{lim}}\left(f_1\right(x)+f_2(x)+....+f_n(x\left)\right)=\underset{x\to a}{\text{lim}}{f}_1\left(x\right)+\underset{x\to a}{\text{lim}}{f}_2\left(x\right)+\cdots +\underset{x\to a}{\text{lim}}{f}_n\left(x\right)$, where $n\in N$ and $\underset{x\to a}{lim}{f}_n\left(x\right)$ exists

• A. Both $I$ and $II$ are individually true and $II$ is the correct explanation of $I$
• B. Both $I$ and $II$ are individually true but $II$ is not the correct explanation of $I$
• C. $I$ is true but $II$ is false
• D. $I$ is false but $II$ is true

Answer 1

The correct option is D $I$ is false but $II$ is true

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

Here Statement $I$ is false and Statement $II$ is true.