A Limit is the value that a graph or a function approaches near a certain point of interest. When we take the limit, we are not concerned about what actually happens a the point of interest. We are only concerned about what happens to the right and to the left of the point of interest. That information alone is enough to determine the limit.
When does the limit exist? When we examine the graph, and the value that to the left and to the right of the point of interest agree, the limit exists. For example, if we have some function and when we approach 2 from the left the value is almost 4 and when we look to the right of 2 height value is almost 4, then the limit from the left and the right agree. We therefore have that the limit as f approaches 2 is 4. This can be denoted lim f-> 2 = 4 If the limit from the left, and the limit from the right do not agree, then the limit does not exist.