Rationalize the numerator by multiplying both numerator and denominator by [itex]\sqrt{n+ a)(n+b)}+ n[/itex]. Then divide both numerator and denominator by n.cauchy21 said:(a) [tex]lim_{n\rightarrow\infty}[/tex] ([tex]\sqrt{(n + a)(n + b)} - n)[/tex] where a, b > 0
A limit of a sequence is the value that the terms of the sequence approach as the index, n, approaches infinity. In other words, it is the value that the terms of the sequence get closer and closer to, but may never actually reach.
To calculate the limit of a sequence, you can use the formula lim n→∞ an = L, where L is the limit and an is the nth term of the sequence. You can also use various techniques such as the squeeze theorem or the ratio test to determine the limit.
If a limit of a sequence does not exist, it means that the terms of the sequence do not approach a single value as the index, n, approaches infinity. This could happen if the terms of the sequence oscillate between two or more values, or if the terms do not approach any value at all.
The parameters a and b affect the limit of a sequence by determining the behavior of the terms of the sequence. For example, if a is a positive number, the terms of the sequence will increase as n increases, while a negative a value will cause the terms to decrease. The parameter b can also shift the entire sequence up or down.
Understanding the limits of sequences is important in mathematics because it allows us to study the behavior of a sequence as the number of terms increases. This can help us make predictions about the behavior of a system or phenomenon, and can also be applied in various fields such as physics, engineering, and economics.