Question

Lim n-->∝ ((2n-5)/(2n-1))^3n-1

Answer 1

Посмотрите предложенное решение; использовался Второй замечательный.

Answer 2

To find the limit of the expression as n approaches infinity, we can use the properties of limits and algebraic manipulations. Let's solve it step by step:

Lim n --> ∞ ((2n - 5) / (2n - 1))^(3n - 1)

First, let's simplify the expression inside the parentheses:

Lim n --> ∞ ((2n - 5) / (2n - 1))

As n approaches infinity, the highest power of n in the numerator and denominator will dominate the expression. We can neglect the lower powers of n:

Lim n --> ∞ (2n / 2n)

Now, divide both the numerator and denominator by n:

Lim n --> ∞ (2 / 2)

Since the limit is just a constant value, it evaluates to 1:

Lim n --> ∞ 1

Now, let's go back to the original expression:

Lim n --> ∞ ((2n - 5) / (2n - 1))^(3n - 1)

We have already simplified the part inside the parentheses to 1. Now, we have:

Lim n --> ∞ 1^(3n - 1)

Any nonzero number raised to the power of zero is equal to 1:

1^0 = 1

So, we have:

Lim n --> ∞ 1

The final result is 1. Therefore, the limit of the expression as n approaches infinity is 1.