CALCULUS 

This branch of mathematics majorly deals with derivatives and integrals. The underlying heart of derivtives is in limits and slope, and of integrals is in areas and summation. Interestingly, most of the development in this area has been due to the inquistiveness of physics scientists. Vectors and Vector calculus is another such area which falls in this category. Also these two concepts are complimentary to each other. Integrals can also be thought as anti-derivatives.

Here is a derivation of derivative of sin(x) function using the fundamental definition of a derivative which uses the concept of limits. The value of these limits are pretty much clear from the graphs shown.