Is it always true? Why?

For example, $\displaystyle f(n) = \sum_{k=1}^n \left(\frac 1{k^2} - \frac 1k \right)$ and $\displaystyle g(n) = \sum_{k=1}^n \frac 1k$ . Since $\displaystyle \lim_{n \to \infty} \sum_{k=1}^n \frac 1k$ does not exist, both $\displaystyle \lim_{n \to \infty} f(x)$ and $\displaystyle \lim_{n \to \infty} g(x)$ do not exist. But the LHS does exist as shown below:

$\begin{aligned} \lim_{n \to \infty} [f(n) + g(n)] & = \lim_{n \to \infty} \left[ \sum_{k=1}^n \left(\frac 1{k^2} - \frac 1k \right) + \sum_{k=1}^n \frac 1k \right] = \lim_{n \to \infty} \sum_{k=1}^n \frac 1{k^2} = \zeta(2) = \frac {\pi^2} 6 \end{aligned}$