This may be too simplistic an example for you, but the best real world example of a limit is the speedometer in your car! The speedometer measures instantaneous velocity, i.e. the velocity right now.. How would you define that? Well, calculus defines it as the limit of the change in distance traveled in a time period as the time period becomes smaller.

Second example: you know how to define the area of a rectangle (a garden plot, a sidewalk)...as the product of the length and width. From that it is a fairly easy step to define the area of a figure "composed" of all straight lines. So you can figure out how much paint to buy to paint any such figure. Now you need to know how much paint to buy to paint a circular water tower. At least one way to define the area of non-rectilinear figures involves the notion of a limit, the limit of the areas on enclosing and enclosed rectangles as one of the dimensions of the rectangle becomes smaller.

Each of these examples involves the fundamental notions of calculus, the derivative and the integral.

And BTW, no "informal" description of a limit will do the job...when you get to calculus you MUST memorize (yes, memorize) the definition of a limit...because if you don't get the definition of a limit correct, your study of calculus will be hopeless!