Please give graphs of different types nonexamples of a (preferably at least one for each type of mistake a student might make trying to draw limits).
I've seen multiple definitions of a limit, and I would like to just see concrete examples of what is not a limit. I am mainly concerned about the uniqueness part and what can violate that. And, does the object that maps to the diagram (in the cones) need to be the terminal object alone or does the mapping from the object from the object to the diagram also need to be terminal. I'm pretty sure the latter, but please give a pitfall example(s).
Working in $\mathbf{Set}$, let $A\odot B$ be the set $$\{\langle i,a,b\rangle\;|\;i\in\{0,1\}\wedge a\in A\wedge b\in B\},$$ and let $\pi_A$ and $\pi_B$ send a member of $A\odot B$ to its second and third component, respectively. Then for any pair of maps $f:C\to A$ and $g:C\to B$, there is a map $h:C\to A\odot B$ with $\pi_Ah=f$ and $\pi_B h=g$; just let $h$ be the function that sends $c\in C$ to $\langle 0,f(c),g(c)\rangle$. But clearly $h$ is not unique with this property; if we chose instead the function that sends $c$ to $\langle 1,f(c),g(c)\rangle$, that would have worked just as well.
I am uncertain what the part of the question regarding terminal objects is asking, but if you can clarify that I'll be happy to expand on this answer.