The intuitive statement

$$\lim_{x\to a}f(x)=L$$

can be made precise without referring to infinitesimals.

For every tolerance $\varepsilon>0$ around the output $L$, there must exist a tolerance $\delta>0$ around the input $a$ such that

$$0<\lvert x-a\rvert<\delta$$

implies

$$\lvert f(x)-L\rvert<\varepsilon$$

The order matters: no matter how small an output tolerance is requested, we must be able to choose an input tolerance that guarantees it. The definition formalizes the idea that $f(x)$ can be forced arbitrarily close to $L$ by taking $x$ sufficiently close to $a$.

Visual intuition: epsilon and delta