Sure, let's evaluate the given limit step-by-step:
[tex]\[
\lim_{x \to \infty} \sqrt{x} (\sqrt{x+3} - \sqrt{x})
\][/tex]
1. Consider the Expression Inside the Limit:
We need to simplify the expression inside the limit: [tex]\(\sqrt{x} (\sqrt{x+3} - \sqrt{x})\)[/tex].
2. Rationalize the Expression Inside the Parentheses:
To facilitate the simplification, we can rationalize the expression inside the parentheses by multiplying and dividing by the conjugate of [tex]\(\sqrt{x+3} - \sqrt{x}\)[/tex], which is [tex]\(\sqrt{x+3} + \sqrt{x}\)[/tex]:
[tex]\[
\sqrt{x} (\sqrt{x+3} - \sqrt{x}) = \sqrt{x} \cdot \frac{(\sqrt{x+3} - \sqrt{x})(\sqrt{x+3} + \sqrt{x})}{\sqrt{x+3} + \sqrt{x}}
\][/tex]
3. Simplify the Numerator:
The numerator becomes a difference of squares:
[tex]\[
(\sqrt{x+3} - \sqrt{x})(\sqrt{x+3} + \sqrt{x}) = (x+3) - x = 3
\][/tex]
So the expression simplifies to:
[tex]\[
\sqrt{x} \cdot \frac{3}{\sqrt{x+3} + \sqrt{x}} = \frac{3\sqrt{x}}{\sqrt{x+3} + \sqrt{x}}
\][/tex]
4. Evaluate the Limit as [tex]\(x\)[/tex] Approaches Infinity:
For large values of [tex]\(x\)[/tex], [tex]\(\sqrt{x+3}\)[/tex] approximately equals [tex]\(\sqrt{x}\)[/tex]. Thus, we can simplify the denominator:
[tex]\[
\sqrt{x+3} + \sqrt{x} \approx \sqrt{x} + \sqrt{x} = 2\sqrt{x}
\][/tex]
Substitute this approximation into the expression:
[tex]\[
\frac{3\sqrt{x}}{\sqrt{x+3} + \sqrt{x}} \approx \frac{3\sqrt{x}}{2\sqrt{x}} = \frac{3}{2}
\][/tex]
Thus, the limit is:
[tex]\[
\lim_{x \to \infty} \sqrt{x} (\sqrt{x+3} - \sqrt{x}) = \frac{3}{2}
\][/tex]
So, the evaluated limit is:
[tex]\[
\boxed{\frac{3}{2}}
\][/tex]
