To solve the limit
[tex]$\lim_{{x \to a}} \frac{\sqrt{2x} - \sqrt{3x - a}}{\sqrt{x} - \sqrt{a}},$[/tex]
let's follow a step-by-step approach.
1. Substitution: Start by substituting [tex]\( x = a \)[/tex] in the limit expression to check if it's an indeterminate form.
[tex]\[ \frac{\sqrt{2a} - \sqrt{3a - a}}{\sqrt{a} - \sqrt{a}} = \frac{\sqrt{2a} - \sqrt{2a}}{0} = \frac{0}{0}. \][/tex]
Since this is an indeterminate form [tex]\( \frac{0}{0} \)[/tex], we'll need to apply some algebraic manipulation to simplify the expression.
2. Manipulate the expression: To simplify the expression, multiply the numerator and the denominator by the conjugate of the denominator:
[tex]\[ \frac{\sqrt{2x} - \sqrt{3x - a}}{\sqrt{x} - \sqrt{a}} \cdot \frac{\sqrt{x} + \sqrt{a}}{\sqrt{x} + \sqrt{a}}. \][/tex]
This results in:
[tex]\[ \frac{(\sqrt{2x} - \sqrt{3x - a})(\sqrt{x} + \sqrt{a})}{(\sqrt{x} - \sqrt{a})(\sqrt{x} + \sqrt{a})}. \][/tex]
The denominator simplifies using the difference of squares formula:
[tex]\[ (\sqrt{x} - \sqrt{a})(\sqrt{x} + \sqrt{a}) = x - a. \][/tex]
Thus, the limit expression becomes:
[tex]\[ \frac{(\sqrt{2x} - \sqrt{3x - a})(\sqrt{x} + \sqrt{a})}{x - a}. \][/tex]
3. Simplify the numerator: Expand the numerator:
[tex]\[ (\sqrt{2x} - \sqrt{3x - a})(\sqrt{x} + \sqrt{a}) = (\sqrt{2x} \cdot \sqrt{x} + \sqrt{2x} \cdot \sqrt{a} - \sqrt{3x - a} \cdot \sqrt{x} - \sqrt{3x - a} \cdot \sqrt{a}). \][/tex]
This simplifies to:
[tex]\[ (\sqrt{2x} \cdot \sqrt{x} - \sqrt{3x - a} \cdot \sqrt{x} + \sqrt{2x} \cdot \sqrt{a} - \sqrt{3x - a} \cdot \sqrt{a}). \][/tex]
Recognizing that [tex]\(\sqrt{x} \cdot \sqrt{x} = x\)[/tex] and similarly for others, the expression becomes:
[tex]\[ (x\sqrt{2} - x\sqrt{3 - a/x} + a\sqrt{2x} - a\sqrt{3x - a}/\sqrt{a}). \][/tex]
To further simplify, we must focus on specific dominant terms as [tex]\(x \rightarrow a\)[/tex].
4. Taking the limit: As [tex]\(x\)[/tex] approaches [tex]\(a\)[/tex], [tex]\(\sqrt{3 - a/x}\)[/tex] becomes constant, and we simplify our expression accordingly by noting:
[tex]\[ \lim_{{x \to a}} \frac{(x\sqrt{2} - x\sqrt{3 - a/x})(\sqrt{x} + \sqrt{a})}{x - a}. \][/tex]
Ultimately, simplifying for when [tex]\(x\)[/tex] trends to [tex]\(a\)[/tex]:
[tex]\[ -\sqrt{2}/2. \][/tex]
Therefore, the limit is
[tex]\[ \boxed{-\sqrt{2}/2}. \][/tex]
