It is correct to say that the concept of limit forms the basis for the study of calculus. It is crucial to understand this concept very well before one starts his study of calculus. A mathematical and equation based treatment of this topic can be found in any standard mathematical text book. Let me start by telling a story.
Once upon a time, in a deep forest, lived a frog. (Usually, in stories, we see frogs living in ponds. But this frog not an aquatic species, it is terrestrial one). After some months, when summer came, a drought was set in the forest, Food supply considerably reduced in the forest. Many animals started to die starving. The hero of the story, Mr.Frog also got struck badly by the drought. He searched for food everywhere but he couldn’t find any. Eventually, he got tired. He lied on the ground, too tired to jump. When he was about to die, a golden sight caught his eye. Lying two feet ahead of him was an insect. It was a dead insect. Mr.Frog tried to stretch his tongue and get the insect, but it was too far away. He tried to jump towards the insect, but he was too tired to jump. Lying on the ground, he thought about his fate. He was about to die starving, with the food right in front of him. At that instant, he started things that people usually do at stated like this-he started to pray. It is evident from our daily life experience that God listens to the prayers of the needy. God himself came to the frog, asking him what he wanted. He could have asked for anything from food to the position of Mr.Lion (king of the jungle, usual stuff). But what he asked surprised even God. He asked God some energy to jump. Now it was God’s turn to surprise the frog. God gave him a boon. God granted him enough energy to jump half the distance to the food. There was no limit in the number of jumps he could make. Even though the intention of God through this boon was to surprise the frog, Mr.Frog was not surprised. He seemed to be extremely happy.
Now let us talk some mathematics. Imagine the Frog and the insect as points on the real number line. Think what would happen when the Frog point covers half the distance to the insect point at every jump. You will see that the frog would never reach the insect! By saying never, I mean Never! (If you don’t see that, just take your time and think. Suppose the distance between the Frog and the insect were four meters. The frog covers two meters in the first jump, one meter in the second jump, half meter in the third jump, and so on. You consider a number of jumps and add the distances, you will not get four. You consider some more jumps, more jumps, more and more jumps; you will not get the number four as the sum. Don’t approximate things, and don’t take into consideration the dimensions of frog and insect. I told you in the very beginning that frog and insect are point objects that are dimensionless.) This is the concept of limit. We have functions in mathematics that are mathematical statements showing how the value of something depends on another. By saying y is a function of x, we mean that the value of y depends on the vale of x in some orderly fashion.
We should get a real value for y for every real value ‘a’ we put to x. We need not put a value of x in every case. We can limit the value of x ‘a’. We can talk about the value of y tending to ‘b’ when the value of x is tending to ‘a’. Remember, it is x tending to ‘a’ and x not equal to ‘a’. The meaning of x tending to ‘a’ is the value of x is approaching ‘a’. It was approaching ‘a’ in the past, it is approaching ‘a’ in the present and it will keep on approaching ‘a’ in the future. (Just like our story where the frog was approaching the insect in the past, it is approaching the insect in the present and it will keep on approaching the insect in the future.) It is a beautiful concept in mathematics. You should understand that the value of x keeps on changing (as it is in an approach process, like the frog). As y depends on x, the value of x continuously changing should imply the value of y correspondingly changing. There comes limit again. In the case of the frog, imagine there was some mud ball on the body of the frog. The statement that the frog approaches food correspondingly implies the mud ball on its body also approaches the food. In the mathematical language, the position of mud ball depends on the position of the frog or the position of mud is a function of the position of the frog.
What is the use of all this? Why all the trouble of limiting functions? There are may uses like checking the continuity of functions, differentiating functions, etc.
Now let us come back to the story. As the frog and the insect are not point objects, the frog did eventually reach the insect when the distance to jump was less than the length of his body. He did obviously enjoy his meal, and had some energy to search for more food and lived till the rain came. The question of the frog using his tongue and getting the food when he was half way is ruled out as God had only given him energy to jump. In fact, it is safe to assume that Mr.Frog obviously knew some mathematics.