To solve $\lim_{x \to \infty} \left( \frac{\sqrt{x} + 2}{x + 3} \right)^x$, let's proceed step-by-step.

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Step 1: Express the function as $e^{\ln}$

The limit can be rewritten using the exponential and natural logarithm: $\lim_{x \to \infty} \left( \frac{\sqrt{x} + 2}{x + 3} \right)^x = \exp\left(\lim_{x \to \infty} x \ln\left(\frac{\sqrt{x} + 2}{x + 3}\right)\right).$

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Step 2: Simplify the logarithmic term

Let: $y = \ln\left(\frac{\sqrt{x} + 2}{x + 3}\right) = \ln(\sqrt{x} + 2) - \ln(x + 3).$

For large $x$, approximate each term:

1. $\ln(\sqrt{x} + 2) = \ln(\sqrt{x}(1 + \frac{2}{\sqrt{x}})) = \ln\sqrt{x} + \ln(1 + \frac{2}{\sqrt{x}})$.
   Using $\ln(1 + u) \approx u$ for small $u$, $\ln(1 + \frac{2}{\sqrt{x}}) \approx \frac{2}{\sqrt{x}}$.
   Thus, $\ln(\sqrt{x} + 2) \approx \frac{1}{2} \ln x + \frac{2}{\sqrt{x}}$.

2. $\ln(x + 3) = \ln(x(1 + \frac{3}{x})) = \ln x + \ln(1 + \frac{3}{x})$.
   Using $\ln(1 + u) \approx u$ for small $u$, $\ln(1 + \frac{3}{x}) \approx \frac{3}{x}$.
   Thus, $\ln(x + 3) \approx \ln x + \frac{3}{x}$.

Now, the difference becomes: $\ln\left(\frac{\sqrt{x} + 2}{x + 3}\right) \approx \left(\frac{1}{2} \ln x + \frac{2}{\sqrt{x}}\right) - \left(\ln x + \frac{3}{x}\right) = -\frac{1}{2} \ln x + \frac{2}{\sqrt{x}} - \frac{3}{x}.$

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Step 3: Multiply by $x$ and analyze the dominant term

$x \ln\left(\frac{\sqrt{x} + 2}{x + 3}\right) \approx x\left(-\frac{1}{2} \ln x + \frac{2}{\sqrt{x}} - \frac{3}{x}\right).$

1. The dominant term here is $-\frac{1}{2} x \ln x$, as it grows faster than the other terms ($\frac{2x}{\sqrt{x}} = 2\sqrt{x}$ and $-3$).
2. Thus, $x \ln\left(\frac{\sqrt{x} + 2}{x + 3}\right) \to -\infty$ as $x \to \infty$.

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Step 4: Conclude the limit

$\exp\left(\lim_{x \to \infty} x \ln\left(\frac{\sqrt{x} + 2}{x + 3}\right)\right) = \exp(-\infty) = 0.$

The limit is: $\boxed{0}.$

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Would you like further clarifications or a breakdown of specific steps?

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Related Questions:

1. What is the growth rate comparison of $\ln x$ and $\sqrt{x}$?
2. How does the expansion of logarithmic terms affect limit problems?
3. Can L'Hôpital's rule be applied to this problem? Why or why not?
4. What are the common dominant term approximations for limits at infinity?
5. How does the natural logarithm transform exponential limits?

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Tip: When analyzing limits involving exponentials, focus on the dominant terms in logarithmic expansions.