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?

The first case works because you have replaced the sum of a varying number of terms with a finite sum of exactly 2 terms, and the replacement is exact with no approximation $$\lim _{m \rightarrow \infty} \left(\frac{1}{2}+\frac{1}{2m} \right)$$
This means it is legitimate to interchange the limit and the sum.
The second case is not correct due to there still being a varying number of terms in the sum. It is not legitimate to apply limits term-by-term in such a case. See the comments for a nice counter-example.
