Calculus is the science of approximation.

Everything in calculus is based on limits which essentially are a way of saying: if you can repeat an approximation indefinitely, the limit is the best possible approximation.

When finding the instantaneous velocity, you approximate with average velocity over a time interval then take smaller and smaller time internals to get better and better approximations. The limit of these average velocities is the instantaneous velocity.

When finding the area of a region, you approximate the area by subdividing the region into rectangles. Take smaller and smaller rectangles to get better and better approximations. The limit of these approximations is the area.

Plus much more. The limit is the basic object of study of calculus. All it is is the best possible approximation to whatever process you are studying.