Let $\mathcal{C}$ is a grothendiect category and consider all of what follows in $\mathcal{C}$.
Let $$\{\varepsilon_i: 0\to A_i \to B_i \to C_i\to 0\ ,\ \phi_i^j\}$$ be a direct system of
short exact sequences.
Is it true to say that $\varepsilon: 0\to \lim A_i \to \lim B_i \to \lim C_i \to 0$ (where $lim$ denotes the direct limit)
is the direct limit $\lim \varepsilon_i$.
Actually we know that there exists a morphism $\varepsilon_i\to \varepsilon$ for each $i$.
What does occur when $\varepsilon_i$ is a split short exact sequence for each $i$? In fact why in this fact the direct limit is a pure short exact sequence?
In a locally finitely presented category, an exact sequence is called pure if it is a direct limit of split short exact sequences. In a locally presentable category the notion of $\alpha$-purity arises in a similar way (In fact pure exact sequences are $\aleph$-pure exact sequences).
For example in the category of modules over a ring, an exact sequence is pure if and only if it is a direct limit of split exact sequences.
Best Answer
I am going to concentrate on the underlying categorical issue here and leave the homological algebra to the experts in that subject.
A direct limit is also known as a colimit. Yours, I take it, is over a sequence $N$, and as such is called directed or filtered.
Being an short exact sequence $0\to A\to B\to C\to 0$ amounts three things:
$A\to B$ is a monomorphism or $0$ is its kernel, which are kinds of limit (or projective limit in old terminology), but finitary ones;
$B\to C$ is an epimorphism, or $C$ is its image, which are other kinds of colimit property, this time finitary ones.
the image of $A\to B$ is the kernel of $B\to C$, which combines properties of both kinds.
Now, limits commute with limits and colimits commute with colimits.
The question is whether filtered colimits commute with finite limits.
Indeed, they do if the category described is by a finitary algebraic theory.
I presume that a Grothendieck category is like this, though Zhen Lin says otherwise.
As David White says, the interest in filtered colimits arises from the fact that they are the ones that commute with finitary stuff such as limits in $\bf Set$.
However, this is not a theorem of categories in general.
As Zhen Lin points out, ${\bf Set}^{op}$ does not have this property. Limits and colimits in $\bf Set$ behave quite differently, as I emphasise in Chapter V of my book, Practical Foundations of Mathematics.
A preorder in which this happens is called meet-continuous. See Counterexamples O 4.5 of A Compendium of Continuous Lattices for complete lattices that are not meet-continuous.
(I had difficulty with my Internet connection when I originally posted this, so some of the intended text got deleted by mistake.)
I repeat that I am not an expert on homological algebra, but I note that the Wikipedia article on Grothendieck categories says explicitly that "direct limits (a.k.a. filtered colimits) of exact sequences are exact".