Question

Lim n → ∞ √ n − √ n − 3

Answer 1

Ответ:

-3

Объяснение:

$\lim_{n \to \infty}(\sqrt{n}-\sqrt{n} ) - \lim_{n \to \infty}(3)$

$\lim_{n \to \infty}(\sqrt{n} ) \\ \lim_{n \to \infty}(\sqrt{n} )$

$+\infty\\+\infty$

$\lim_{n \to \infty}(\sqrt{n}-\sqrt{n} )$

$\lim_{n \to \infty}(0 )$

$0-3=-3$

Answer 2

To find the limit of the given expression as n approaches infinity:

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lim (n → ∞) (√n - √(n - 3))

Let's simplify the expression first by using the difference of squares formula: $a^2 - b^2 = (a + b)(a - b)$.

In this case, let $a = \sqrt{n}$ and $b = \sqrt{n - 3}$. Applying the difference of squares formula:

$\sqrt{n} - \sqrt{n - 3} = (\sqrt{n} + \sqrt{n - 3})(\sqrt{n} - \sqrt{n - 3})$.

Now the expression becomes:

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lim (n → ∞) ((√n + √n - √(n - 3))(√n - √(n - 3)))

Simplify the terms inside the parentheses:

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lim (n → ∞) (2√n - √(n - 3))(√n - √(n - 3))

As $n$ approaches infinity, $\sqrt{n}$ grows much faster than $\sqrt{n - 3}$, so we can consider only the dominant terms:

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lim (n → ∞) (2√n * √n)

Simplify further:

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lim (n → ∞) 2n

As $n$ approaches infinity, the limit of $2n$ is also infinity:

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lim (n → ∞) 2n = ∞

Therefore, the limit of the given expression as $n$ approaches infinity is infinity.