As a subject matter expert in mathematical analysis, I can provide a detailed explanation on the concepts of "undefined" and "does not exist" within the context of limits in calculus. These two terms are often used interchangeably in colloquial language, but in a technical and mathematical sense, they have nuanced differences that are important to understand. When we discuss limits in calculus, we are referring to the behavior of a function as its input approaches a certain value. A limit is said to be undefined if the function's behavior as it approaches a certain point is not well-defined or cannot be determined within the framework of standard mathematical operations. This can happen for a variety of reasons, such as the function not being continuous at that point, or the function not being defined at the point in question. On the other hand, when we say that a limit does not exist, we are making a stronger statement. It implies that not only is the limit not well-defined, but there is also no value to which the function can approach as the input gets arbitrarily close to the point of interest. In other words, the function's values do not settle down to a single number or infinity as the input approaches the point. Let's consider an example to illustrate the difference. Take the function f(x) = 1/x. As x approaches zero from the right (positive values), the function values get larger and larger, tending toward infinity. Similarly, as x approaches zero from the left (negative values), the function values become more and more negative, also tending toward negative infinity. In this case, we say that the limit of f(x) as x approaches zero does not exist because the function's values do not approach a single value or infinity from both sides of zero. However, if we were to consider a function that is not defined at a certain point, such as f(x) = √(x) at x = -1, we might say that the limit is undefined at that point because the function is not defined for negative values of x. The square root function is not meaningful for negative inputs within the set of real numbers, so we cannot compute a limit there. It's important to note that the phrase "the limit is undefined" can sometimes be used to mean that the limit does not exist, especially in informal mathematical discussions. However, the technical definition of a limit requires that we distinguish between the two. A limit is undefined when it cannot be computed or does not fit within the standard mathematical framework, while it does not exist when there is no value that the function approaches as the input approaches the point in question. In summary, while "undefined" and "does not exist" can be used to describe similar situations, they are not synonymous in a strict mathematical sense. The term "undefined" speaks to the limitations of our mathematical tools or the function's domain, whereas "does not exist" speaks to the behavior of the function's values as they approach a certain point without settling on a specific limit. read more >>

The answer to your question is that the limit is undefined if the limit does not exist as described by this technical definition. ... In this example the limit of f(x), as x approaches zero, does not exist since, as x approaches zero, the values of the function get large without bound.read more >>