Evaluating $\lim _{m\to\infty}\left(\frac1{m^2}+\frac2{m^2}+\frac3{m^2}+\cdots+\frac{m}{m^2}\right)$. Where's my error? Question is following
$$\lim _{m \to \infty}\left(\frac{1}{m^{2}}+\frac{2}{m^{2}}+\frac{3}{m^{2}}+\cdots+\frac{m}{m^{2}}\right)$$
method-1
$$\lim _{m \rightarrow \infty}\left(\frac{m(m+1)}{2 m^{2}}\right)$$
$$=\frac{1}{2}$$
method-2(applying limits individually)
$$\lim _{m \rightarrow \infty} \frac{1}{m^{2}}+\lim _{m \rightarrow \infty} \frac{2}{m^{2}}+\lim _{m \rightarrow \infty} \frac{3}{m^{2}}+\infty=0$$
Since denominator is always greater than the numerator
What’s wrong with method-2. Is it wrong to apply limits individually. If so then how?

First case is correct. Second case wrong: limit of sums equal sum of limits  when we have fixed amount of summands.
