When $\mathcal{J}$ is discrete with objects $\{1, \ldots, n\}$, a diagram is a family of objects $A_1, \ldots, A_n$, a cone is a family of maps from a single object to each $A_i$, and the limit is the product $A_1 \times \cdots \times A_n$.

In $\mathbf{Ab}$, the equalizer of $f, g : A \to B$ is the kernel of $f - g$: $\mathrm{eq}(f,g) = \ker(f - g) = \{a \in A : f(a) = g(a)\}$.

The inverse limit $\varprojlim \mathbb{Z}/p^n\mathbb{Z}$ (over the system $\cdots \to \mathbb{Z}/p^3\mathbb{Z} \to \mathbb{Z}/p^2\mathbb{Z} \to \mathbb{Z}/p\mathbb{Z}$) is the $p$-adic integers $\mathbb{Z}_p$. This is a filtered inverse limit in $\mathbf{Ring}$.

The direct limit $\varinjlim \mathbb{Z}/n!\mathbb{Z}$ over the inclusions gives the group $\mathbb{Q}/\mathbb{Z}$. More generally, every module is a filtered colimit of its finitely generated submodules.

The coequalizer of $f, g : A \rightrightarrows B$ in $\mathbf{Set}$ is the quotient $B / {\sim}$ where $\sim$ is the equivalence relation generated by $f(a) \sim g(a)$ for all $a \in A$.

In $\mathbf{Grp}$, the coequalizer of $f, g : H \rightrightarrows G$ is the quotient $G / N$ where $N$ is the normal subgroup generated by $\{f(h) \cdot g(h)^{-1} : h \in H\}$.

For a sheaf $\mathcal{F}$ on a topological space $X$, the stalk at $x \in X$ is the filtered colimit $\mathcal{F}_x = \varinjlim_{U \ni x} \mathcal{F}(U)$ over all open neighborhoods of $x$.

In the functor category $[\mathcal{C}, \mathcal{D}]$, limits are computed pointwise: if $\mathcal{D}$ has all limits of shape $\mathcal{J}$, then $(\varprojlim F_i)(A) = \varprojlim F_i(A)$ for each $A \in \mathcal{C}$. This is why presheaf categories $[\mathcal{C}^{\mathrm{op}}, \mathbf{Set}]$ have all limits and colimits.

The limit $\varprojlim D$ can be described as the equalizer of two maps between products: $\varprojlim D = \mathrm{eq}\left(\prod_{j} D(j) \rightrightarrows \prod_{u : j \to j'} D(j')\right)$. This gives the construction of limits via products and equalizers.

The profinite completion $\hat{G}$ of a group $G$ is the inverse limit $\hat{G} = \varprojlim_{N \trianglelefteq G,\, [G:N] < \infty} G/N$. The profinite completion of $\mathbb{Z}$ is $\hat{\mathbb{Z}} = \prod_p \mathbb{Z}_p$.

Any group $G$ is a filtered colimit of its finitely generated subgroups. Any $R$-module is a filtered colimit of its finitely generated submodules. Any ring is a filtered colimit of its finitely generated subrings.

In $\mathbf{Top}$, limits are formed by taking the limit of the underlying sets (computed in $\mathbf{Set}$) and equipping it with the initial topology (coarsest making all projections continuous). Colimits use the final topology.

The categories $\mathbf{Set}$, $\mathbf{Grp}$, $\mathbf{Ab}$, $R\text{-}\mathbf{Mod}$, $\mathbf{Top}$, and $[\mathcal{C}, \mathcal{D}]$ (when $\mathcal{D}$ is complete/cocomplete) are all complete and cocomplete.