IMO you are wondering, how limits in analysis relate to limits in category theory. AFAIK they are not related — simple. Instead, limits in category theory are a generalization of the greatest lower bound in a preorder.
At elementary level limits are defined for “diagrams” (you should know what are they). But every functor can be interpreted as a diagram. So people just talk about limits for functors. The functor $F:J\to A$ is interpreted as the diagram in the category $A$. $J$ is the category called a “scheme” of the diagram $F$. E.g. the scheme of binary product diagrams is the free category over the graph consisting of 2 vertexes and no edges.
It is important that functors map not only objects and morphisms, but also diagrams. This is obvious, because a diagram is a bunch of objects and morphisms. $V:A\to X$ maps $F$ to $V(F)$. $F, V(F)$ have the same scheme. Morphisms in $F$ commute, $F$ preserves compositions of morphisms, then morphisms in $V(F)$ commute, then $V(F)$ is a diagram. If you look at $F$ as a functor, then $V(F)$ is actually $V\circ F$.
“$V:A\to X$ creates limits” means that for every diagram $F$ in $A$ and every limit $\tau$ of $V(F)$ there exists a unique limit of $F$ which is a preimage (along $V$) of $\tau$. E.g. if we choose a particular limit $F$ — the binary product diagram of some groups $G_0, G_1$, we choose a particular functor $V$ — the forgetful functor $Grp\to Set$, and $V(G_0)\times V(G_1)$ — the Cartesian product of the carriers (underlying sets) of those groups — is the limit of $V(F)$, then there exists a unique group with the carrier $V(G_0)\times V(G_1)$ (it has this carrier because it is the preimage of the set $V(G_0)\times V(G_1)$), and and that group is the categorical product of $G_0, G_1$ with projections $\pi_0:V(G_0)\times V(G_1)\to V(G_0), \pi_1:V(G_0)\times V(G_1)\to V(G_1)$ understood as group homomorphisms. I.e. that a Cartesian product of carriers can be extended to a categorical product of groups, which is called also a direct product of groups.