To evaluate the limit [tex]\(\lim_{x \to a} \frac{\sqrt{x + a} - \sqrt{3x - a}}{x - a}\)[/tex], we will proceed with a detailed, step-by-step approach.
1. Recognize the Indeterminate Form:
Since direct substitution of [tex]\(x = a\)[/tex] gives:
[tex]\[
\frac{\sqrt{a + a} - \sqrt{3a - a}}{a - a} = \frac{\sqrt{2a} - \sqrt{2a}}{0} = \frac{0}{0},
\][/tex]
we have an indeterminate form of type [tex]\(\frac{0}{0}\)[/tex].
2. Use Algebraic Manipulation to Simplify:
To simplify the given expression, consider rationalizing the numerator. Multiply the numerator and the denominator by the conjugate of the numerator:
[tex]\[
\frac{\sqrt{x + a} - \sqrt{3x - a}}{x - a} \cdot \frac{\sqrt{x + a} + \sqrt{3x - a}}{\sqrt{x + a} + \sqrt{3x - a}}.
\][/tex]
This multiplication leads to:
[tex]\[
\frac{(\sqrt{x + a})^2 - (\sqrt{3x - a})^2}{(x - a)(\sqrt{x + a} + \sqrt{3x - a})}.
\][/tex]
Simplifying the numerator:
[tex]\[
(\sqrt{x + a})^2 - (\sqrt{3x - a})^2 = (x + a) - (3x - a) = x + a - 3x + a = -2x + 2a = 2(a - x).
\][/tex]
Hence, the expression becomes:
[tex]\[
\frac{2(a - x)}{(x - a)(\sqrt{x + a} + \sqrt{3x - a})}.
\][/tex]
3. Simplify the Fraction:
Recognize that [tex]\(a - x = -(x - a)\)[/tex], so:
[tex]\[
\frac{2(a - x)}{(x - a)(\sqrt{x + a} + \sqrt{3x - a})} = \frac{2(-1)(x - a)}{(x - a)(\sqrt{x + a} + \sqrt{3x - a})} = \frac{-2(x - a)}{(x - a)(\sqrt{x + a} + \sqrt{3x - a})}.
\][/tex]
We can cancel out [tex]\((x - a)\)[/tex] in the numerator and the denominator:
[tex]\[
\frac{-2}{\sqrt{x + a} + \sqrt{3x - a}}.
\][/tex]
4. Evaluate the Limit:
Now, taking the limit as [tex]\(x \to a\)[/tex]:
[tex]\[
\lim_{x \to a} \frac{-2}{\sqrt{x + a} + \sqrt{3x - a}} = \frac{-2}{\sqrt{a + a} + \sqrt{3a - a}} = \frac{-2}{\sqrt{2a} + \sqrt{2a}} = \frac{-2}{2\sqrt{2a}} = \frac{-1}{\sqrt{2a}}.
\][/tex]
5. Simplify the Final Result:
Simplifying further, we get:
[tex]\[
\lim_{x \to a} \frac{\sqrt{x + a} - \sqrt{3x - a}}{x - a} = \frac{-1}{\sqrt{2a}}.
\][/tex]
Thus, the limit is:
[tex]\[
\boxed{-\frac{\sqrt{2}}{2\sqrt{a}}}.
\][/tex]
