The problem involves evaluating the limit as $x \to \infty$ of the given expression:

$\lim_{x \to \infty} \frac{\sqrt{x} + \sqrt{x}}{\sqrt{2x}}$

Let's simplify the expression step by step:

1. Combine terms in the numerator: $\sqrt{x} + \sqrt{x} = 2\sqrt{x}.$

   So the expression becomes: $\frac{2\sqrt{x}}{\sqrt{2x}}.$

2. Simplify the fraction: $\frac{2\sqrt{x}}{\sqrt{2x}} = 2 \cdot \frac{\sqrt{x}}{\sqrt{2x}}.$

3. Simplify the square root ratio: $\frac{\sqrt{x}}{\sqrt{2x}} = \frac{\sqrt{x}}{\sqrt{2} \cdot \sqrt{x}} = \frac{1}{\sqrt{2}}.$

4. Multiply: $2 \cdot \frac{1}{\sqrt{2}} = \frac{2}{\sqrt{2}}.$

5. Simplify further: $\frac{2}{\sqrt{2}} = \sqrt{2}.$

Final Answer:

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

Let me know if you want additional clarification or have questions. Here are some related topics you might want to explore:

1. How to evaluate other limits involving square roots and infinity?
2. How to handle indeterminate forms in limits?
3. The role of simplifying fractions in limit problems.
4. Methods for rationalizing square root terms.
5. Examples of limits at infinity in different contexts.

Tip: When dealing with limits involving square roots, factor out the dominant term for simplification.