Question

For any sequence (an) of real numbers,

1) lim UN as N goes to infinity exists and is defined by ____.
2) lim VN as N goes to infinity exists and is defined by ___.

Answer 1

In mathematics, the limits of sequences (UN) and (VN) as N approaches infinity exist if they converge to specific real numbers, which is the value that the sequences approach.

Step-by-step explanation:

The question seems to be referring to the definition and properties of limits in mathematics, particularly involving sequences of real numbers.

1) The limit of a sequence (UN) as N approaches infinity exists if the sequence converges to a specific real number. This limit would be considered the value that UN approaches as N becomes very large. Often, this limit, if it exists, is denoted as L and is the value that the members of the sequence get arbitrarily close to as N goes to infinity.

2) Similarly, the limit of a sequence (VN) as N approaches infinity exists if there is a specific real number that the sequence converges to.

The concept of limits is significant in various fields of mathematics and physics. For example, understanding the behavior of a function as it approaches an asymptote is crucial when dealing with real-world scenarios, such as calculating the potential energy of a system in physics or determining normal distribution probabilities in statistics.