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.