Consider the limit
$$\lim\limits_{x\to a}\ (x+t^3)=l$$
This means $$\forall \epsilon>0,\exists \delta>0\ s.t.|x-a|<\delta \implies |x+t^3-l|<\epsilon\tag{1}$$
Assume we don't have prior knowledge of theorems about limits, we only know the above definition of what a limit is. In other words, assume this is a chapter in a book, about limits, and the only information presented so far is the definition of limits (this is the case for ch. 5 of Spivak's Calculus).
If we have a guess of what the limit $l$ is, we can verify if it is indeed the limit by subbing it into $(1)$. For example, if we guess $l=a+t^3$ then
$$|x-a|<\delta \implies |x+t^3-a-t^3|=|x-a|<\epsilon$$
$\forall \epsilon$ we can choose $\delta=\epsilon$ to make the statement above true always, proving that $l=a+t^3$ is true.
If we don't know what $l$ is, how can we find it without guessing a value?