Real-life limits are used any time you have some type of real-world application approach a steady-state solution.
As an example, we could have a chemical reaction in a beaker start with two chemicals that form a new compound over time. The amount of the new compound is the limit of a function as time approaches infinity.
Similarly, if you drop an ice cube in a glass of warm water and measure the temperature with time, the temperature eventually approaches the room temperature where the glass is stored. Measuring the temperature is a limit again as time approaches infinity.
Limits are also used as real-life approximations to calculating derivatives. It is very difficult to calculate a derivative of complicated motions in real-life situations. So, to make calculations, engineers will approximate a function using small differences in the a function and then try and calculate the derivative of the function by having smaller and smaller spacing in the function sample intervals. For example, when designing the engine of a new car, an engineer may model the gasoline through the car's engine with small intervals called a mesh, since the geometry of the engine is too complicated to get exactly with simply functions such as polynomials. These approximations always use limits.