To solve the limit [tex]\(\lim _{x \rightarrow a} \frac{\sqrt{3a-x}-\sqrt{x+a}}{4(x-a)}\)[/tex], we can proceed with the following steps:
1. Identify the Indeterminate Form:
Substitute [tex]\(x = a\)[/tex] in the given expression:
[tex]\[
\frac{\sqrt{3a-a} - \sqrt{a+a}}{4(a-a)} = \frac{\sqrt{2a} - \sqrt{2a}}{0} = \frac{0}{0}
\][/tex]
This is an indeterminate form, so we need to apply a different method to evaluate the limit.
2. Rationalize the Numerator:
To deal with the indeterminate form, we rationalize the numerator by multiplying and dividing by the conjugate of the numerator:
[tex]\[
\frac{\sqrt{3 a - x} - \sqrt{x + a}}{4(x - a)} \cdot \frac{\sqrt{3 a - x} + \sqrt{x + a}}{\sqrt{3 a - x} + \sqrt{x + a}}
\][/tex]
3. Simplify the Expression:
The product of the numerator and its conjugate simplifies using the difference of squares:
[tex]\[
(\sqrt{3 a - x} - \sqrt{x + a})(\sqrt{3 a - x} + \sqrt{x + a}) = (3 a - x) - (x + a) = 2 a - 2 x
\][/tex]
So the expression becomes:
[tex]\[
\frac{(2 a - 2 x)}{4(x - a)(\sqrt{3 a - x} + \sqrt{x + a})}
\][/tex]
4. Factor and Simplify Further:
Factor out a -2 from the numerator:
[tex]\[
\frac{-2(x - a)}{4(x - a)(\sqrt{3 a - x} + \sqrt{x + a})}
\][/tex]
Cancel out the common [tex]\((x - a)\)[/tex] term from the numerator and the denominator:
[tex]\[
\frac{-2}{4(\sqrt{3 a - x} + \sqrt{x + a})} = \frac{-1}{2(\sqrt{3 a - x} + \sqrt{x + a})}
\][/tex]
5. Evaluate the Limit:
Now compute the limit as [tex]\(x\)[/tex] approaches [tex]\(a\)[/tex]:
[tex]\[
\lim_{x \to a} \frac{-1}{2(\sqrt{3 a - x} + \sqrt{x + a})} = \frac{-1}{2(\sqrt{3 a - a} + \sqrt{a + a})} = \frac{-1}{2(\sqrt{2a})}
\][/tex]
Simplify:
[tex]\[
\frac{-1}{2(\sqrt{2a})} = \frac{-1}{2\sqrt{2a}} = -\frac{1}{2\sqrt{2a}}
\][/tex]
6. Final Adjustment and Coefficients:
There is a 4 in the denominator from the original expression, consider it:
[tex]\[
= \frac{-1}{4\sqrt{2a}} = -\frac{1}{4 \cdot 2\sqrt{a}} = -\frac{1}{8\sqrt{a}}
\][/tex]
Therefore, the final result is:
[tex]\[
\lim _{x \rightarrow a} \frac{\sqrt{3a-x}-\sqrt{x+a}}{4(x-a)} = -\frac{\sqrt{2}}{8\sqrt{a}}
\][/tex]
