The rigorous foundation of calculus rests on the precise definition of a limit.
This is often the first major hurdle in analysis: quantifying "closeness".
We say $\lim_{x \to c} f(x) = L$ if:
if $0 < |x - c| < \delta$, then $|f(x) - L| < \epsilon$.
Think of this as a game: Player A (the Skeptic) picks a narrow target band around the limit ($\epsilon$). Player B (the Prover) must find a width ($\delta$) around $c$ such that the function graph stays completely inside the $\epsilon$-band.
If you can always find a working $\delta$ no matter how small $\epsilon$ gets, the limit exists! Notice how for steeper functions, your $\delta$ must be smaller to keep the graph "tamed" within the box.