Amit M Agarwal Solutions for Chapter: Limits, Exercise 6: Target Exercise 5.6

The value of $\underset{n\to \infty }{\lim }\left(\frac{[1^{2}x+1^2]+[2^{2}x+2^2]+...+[n^{2}x+n^2]}{n^3}\right)$ is: (where $\left[·\right]$ denotes the greatest integer function.)

The value of $\underset{x\to \frac{\pi }{2}}{\lim }\left\{\underset{n\to \infty }{\lim }\frac{\left[1^2(\sin x{)}^x\right]+\left[2^2(\sin x{)}^x\right]+...+\left[n^2(\sin x{)}^x\right]}{n^3}\right\}$ is (where $\left[·\right]$ denotes the greatest integer function.):

Remark: Limit for $x$ is changed from $\infty$ to $\frac{\pi }{2}$ else the limit will not exist.

The value of $\underset{x\to 1^{+}}{\lim }\frac{{\int }_1^x\left|t-1\right|dt}{\sin \left(x-1\right)}$ is?

The value of $\underset{n\to \infty }{\lim } \sum _{k=1}^n \log {\left(1+\frac{k}{n}\right)}^{\frac{1}{n}}$, is

The value of $\underset{n\to \infty }{\lim }\left(\frac{1}{na}+\frac{{\displaystyle 1}}{{\displaystyle na+1}}+\frac{{\displaystyle 1}}{{\displaystyle na+2}}+....+\frac{{\displaystyle 1}}{{\displaystyle nb}}\right),$ is