The limit of a function at a given point tells what values are being approached or what values is the function moving toward; in other words the limit tells how the function behaves around a given point. It's also possible to move toward a point without ever actually reaching the point. Suppose that you have a rope that is one meter long. You divide the rope in half and then take one of the half pieces. So, you now have a rope 0.5 meters long. You next divide that piece in half, and take one of the half pieces. You now have a piece of rope that is 0.25 meters long. You in half divide it again, and take another half piece which will be 0.125 meters long. Observe that this process can be represented by a sequence of numbers, If the dividing process is continued indefinitely then what is the smallest possible value that can be obtained for the length of rope? What value is being approached for the length of the rope as you continue to cut it in half? Another way of saying this is what value is being approached by the sequence of numbers?

We call this value the limit or the limiting value of the process.

Now, suppose that you start with a bowl containing one red marble and two blue marbles. Thus, the ratio of red to blue marbles in one to two or 1:2. Next, you add one red marble and one blue marble to the bowl, so that there are now two red marbles and three blue marbles for a ratio of 2:3. You continue by adding another red marble and another blue marble to get a ratio of three red to four blue marbles in the bowl or 3:4. Suppose you continue this process indefinitely. What would be the limiting value of the ratio of red to blue marbles? This situation can be represented by the sequence of numbers below.