Question

Как решить lim корень из 2-x-корень из 2плюс x разделить на x в квадрате плюс 2x

Answer 1

Решение во вложении------------------------------

Answer 2

Solving the Limit

To solve the limit of the given expression, we can use the following steps:

1. Identify the Limit Expression: The given expression is $\lim_{{x \to \infty}} \frac{\sqrt{2-x\sqrt{2}+x}}{x^2+2x}$.

2. Simplify the Expression: We can simplify the expression by rationalizing the numerator and factoring the denominator.

3. Apply the Limit: After simplifying the expression, we can then apply the limit as $x$ approaches infinity to find the solution.

Let's proceed with these steps.

Simplification and Solution

First, let's simplify the given expression:

The given expression is $\lim_{{x \to \infty}} \frac{\sqrt{2-x\sqrt{2}+x}}{x^2+2x}$.

We can rationalize the numerator by multiplying both the numerator and the denominator by the conjugate of the numerator, which is $\sqrt{2-x\sqrt{2}+x} + \sqrt{2}$.

This gives us: $\lim_{{x \to \infty}} \frac{\sqrt{2-x\sqrt{2}+x} \cdot (\sqrt{2-x\sqrt{2}+x} + \sqrt{2})}{(x^2+2x) \cdot (\sqrt{2-x\sqrt{2}+x} + \sqrt{2})}$.

After simplifying, we get: $\lim_{{x \to \infty}} \frac{2-x\sqrt{2}+x+2\sqrt{2}}{x^2\sqrt{2}+2x\sqrt{2}+x\sqrt{2}+2x^2+4x}$.

Now, we can apply the limit as $x$ approaches infinity to find the solution.

Let's calculate the limit and provide the final solution.

Final Solution

After applying the limit as $x$ approaches infinity, the solution to the given limit expression is:

$\lim_{{x \to \infty}} \frac{\sqrt{2-x\sqrt{2}+x}}{x^2+2x} = 0$.

This means that as $x$ approaches infinity, the value of the given expression approaches 0.

I hope this helps! If you have any further questions or need additional assistance, feel free to ask.