Perhaps this is beyond your curriculum, but one could also use the Stolz-Cesàro theorem.

$$\lim_{n\to\infty}\frac{1+2+\cdots+(n-1)+n}{n^2}$$

Denote the numerator and denominator, respectively, by

$$a_n=\sum_{k=1}^nk$$
$$b_n=n^2$$

It's easy to show that $b_n$ is strictly monotone and divergent. Stolz-Cesàro then says that if

$$\lim_{n\to\infty}\frac{a_{n+1}-a_n}{b_{n+1}-b_n}$$

exists, then it is equal to

$$\lim_{n\to\infty}\frac{a_n}{b_n}$$

We have

$$\lim_{n\to\infty}\frac{a_{n+1}-a_n}{b_{n+1}-b_n}=\lim_{n\to\infty}\frac{\sum\limits_{k=1}^{n+1}k-\sum\limits_{k=1}^nk}{(n+1)^2-n^2}=\lim_{n\to\infty}\frac{n+1}{2n+1}=\frac12$$

As the Wiki page mentions, the theorem is sometimes referred to as "L'Hopital's rule for sequences". If you find that intuitive (related question), then perhaps you will think the same of the approach using this theorem.