The value of a limit is the value that a function or sequence approaches as its input or index approaches a certain value. It can be thought of as the "end result" of a mathematical process.
The value of a limit is important because it allows us to understand the behavior of a function or sequence at certain points, even if those points are not defined. It also helps us make predictions and solve problems in calculus and other areas of mathematics.
The value of a limit is calculated by evaluating the function or sequence at values that are closer and closer to the desired point, and observing the trend of these values. If the values approach a single, finite number, then that number is the value of the limit.
Yes, the value of a limit can be different from the value of a function at that point. This can occur when the function has a discontinuity or a jump at that point, or when the function is undefined at that point.
If the value of a limit does not exist, it means that the function or sequence does not approach a finite number as its input or index approaches a certain value. This could be due to oscillation, divergence, or other factors, and further analysis is needed to determine the behavior of the function at that point.