To prove that a limit doesn't exist, you need to show that there are two different possible limits for the same function at a specific point. This can be done by approaching the point from different directions and showing that the function has different values or does not approach a single value.
Some common techniques used to prove a limit doesn't exist include using the Squeeze Theorem, the definition of a limit, and the properties of limits. These techniques involve analyzing the behavior of the function as it approaches the point in question.
Yes, a limit can fail to exist at a finite value. This can occur if the function has a vertical asymptote or a jump discontinuity at the point in question. In these cases, the limit will approach either positive or negative infinity, rather than a finite value.
Yes, a limit can fail to exist at infinity. This can happen if the function oscillates or has a behavior that does not approach a single value as the input approaches infinity. In these cases, the limit is said to be undefined or does not exist.
Proving that a limit doesn't exist is essential for understanding the behavior of a function and making accurate predictions about its values. It also helps to identify points of discontinuity and asymptotes, which are important in various areas of mathematics and science.