Final answer:
An improper integral ∫[a to b] f(x)dx is considered 'divergent' if the corresponding limit does not exist, which identifies whether the integral approaches a finite value or not. the correct answer to the question is B) Divergent.
Step-by-step explanation:
The question involves the concept of improper integrals in mathematics, particularly in the field of calculus. When considering the improper integral ∫[a to b] f(x)dx, we are looking at the integral of a function f(x) over the interval from a to b, where either a or b (or both) may be infinity, or the function f(x) may have an infinite discontinuity within the interval (a, b). If the corresponding limit of the integral as x approaches the bounds of the interval does not exist, the integral is said to be divergent.
One way to think about improper integrals is to consider the area under a curve f(x) from a point x1 to another point x2, as shown in Figure 7.8. If you integrate from a finite value a to an infinite value b, or if the function f(x) goes to infinity at some point in the interval, this is categorized as an improper integral. In such cases, you look at the limit of the integral as x approaches the infinity or the point of discontinuity. If this limit exists and is a finite number, you have a convergent integral. If this limit does not exist or is infinite, then the integral is divergent.
Therefore, the correct answer to the question is B) Divergent. Given the context of the question and the corresponding limit for the improper integral, we can determine that if the limit does not exist, the proper terminology to describe the integral is 'divergent.'