Look at a concrete example of a limit. Consider $$\lim_{x \to 1} \frac{x^2-1}{x-1}$$
This is an easy limit to figure out, since for all $x \ne 1$, $$\frac{x^2-1}{x-1} = x+1$$
Clearly the limit should be the value of $x + 1$ when $x = 1$. That is, $2$. But the fraction is not defined at $x = 1$, where the limit is being taken.
For any value of $x$ near but not equal to $1$, the value of the fraction is near, but not equal to $2$. You might put $x = 0.9$ or $0.99$ or $0.999$ or ... into the fraction to get values of $1.9$ or $1.99$, or $1.999$ or .... These are approximations to the limit. But by its definition the limit itself is not an approximation. Instead, it is that value that is being approximated. Since the closer the input $x$ is to $1$, the closer the output value is to $2$. The limit is not any of the approximations. It is instead $2$ itself, a number you cannot get out of the fraction, since $x$ cannot be $1$.
Limits do not "approach some value". The function is what approaches the value. The limit is the value being approached. Taking a limit can be thought of as a process. But the limit itself is not the process, but the result you get from the process.
This concept does not need to involve infinities or infinitesimals. The standard definition of $$\lim_{x \to a} f(x)$$ is that it is the unique number $L$ which satisfies the condition
For every $\epsilon > 0$, there is some $\delta > 0$ such that for all $x$ with $0<|x - a| < \delta$, we have $|f(x) - L| < \epsilon$.
No mention of infinity or infinitesimals or processes in that definition. The limit is just some number $L$ that meets this condition. If there is no such number, we say the limit does not converge. If there is such an $L$, then that is the limit (you can prove from the condition that there can only be one such number).
Derivatives are limits. If $dx \equiv x-a$ is finite, no matter how small, then in general you do not get the derivative. All you get is an approximation. The derivative is the value those approximations are approximating.
