What is an intuitive approach to solving $\lim_{n\rightarrow\infty}\biggl(\frac{1}{n^2} + \frac{2}{n^2} + \frac{3}{n^2}+\dots+\frac{n}{n^2}\biggr)$?

Intuition should say:

the denominator grows with $n^2$, the numerator grows with $n$. However, the number of fractions also grows by $n$, so the total growth of the numerator is about $n^2$.

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And that's where intuition stops. From here on, you go with logic and rigor, not intuition.

And it gets you to

$$\frac1{n^2} + \frac2{n^2}+\cdots + \frac{n}{n^2} = \frac{1+2+3+\cdots + n}{n^2} = \frac{\frac{n(n+1)}{2}}{n^2} = \frac{n^2+n}{2n^2}$$

and you find that the limit is $\frac12$ (not $1$!)

With integrals: $\lim_{n\rightarrow\infty}\biggl(\frac{1}{n^2} + \frac{2}{n^2} + \frac{3}{n^2}+\dots+\frac{n}{n^2}\biggr)=\lim_{n\rightarrow\infty} \frac{1}{n}\biggl(\frac{1}{n} + \frac{2}{n} + \frac{3}{n}+\dots+\frac{n}{n}\biggr)= \int_{0}^1 x dx =\frac{1}{2}$.

Notice that:

$$\lim_{n\rightarrow\infty}\dfrac{1}{n^2} + \dfrac{2}{n^2} + \dfrac{3}{n^2}+\cdots+\dfrac{n}{n^2}=\lim_{n\rightarrow\infty} \frac{1}{n^2}\left(1+2+\dots n\right)$$ and this last sum can be replaced by the Gauss formula, so it becomes: $$\lim_{n\rightarrow\infty}\frac{1}{n^2}\left(\frac{n(n+1)}{2}\right)$$

$$=\lim_{n\rightarrow\infty}\frac{n+1}{2n}=\frac{1}{2}\left(\lim_{n\rightarrow\infty} \frac{n+1}{n}\right)=\frac{1}{2}\left(\lim_{n\rightarrow\infty} 1+\frac{1}{n}\right)=\frac{1}{2}$$