Question

$\underset{n\to \infty }{\lim }\frac{n^2}{1+2+3+...+n}$

Answer 1

$$\begin{gathered} \underset{n\to \infty }{\lim }\left[\frac{n^2}{1+2+3.....n}\right] \\ \mathrm{It}\mathrm{is}\mathrm{of}\mathrm{the}\mathrm{form}\frac{\infty }{\infty }. \\ \Rightarrow \underset{n\to \infty }{\lim }\left[\frac{n^2}{n{\displaystyle \frac{\left(n+1\right)}{2}}}\right] \\ =\underset{n\to \infty }{\lim }\left[\frac{2n}{n+1}\right] \\ \text{Dividing the numerator and the denominator by n:} \\ \underset{n\to \infty }{\lim }\frac{2}{1+{\displaystyle \frac{1}{n}}} \\ =2 \end{gathered}$$