I'm a teacher in electronics department, I did a light basics of limits, differentiation and integration last semester in just solving problems.

But I want a deep understanding of what these topics actually are? Where can I use limits? Is limits still valuable these days?

What is differentiation? I know it's useful in drawing the actual graph of a certain function. For example, the movement of a running man. Scientists can develop a mathematical function that approximates the movement of the running man, but these functions are actually the integration of this movement! Is it? This is my guess. And the differentiation is the functions that draw how the man actually is running, then the original integration functions are the final results of each movement for the running man.

How about this example?