To solve the limit [tex]\(\lim_{{x \to a}} \frac{\sqrt{x + a} - \sqrt{3x - 9}}{x - a}\)[/tex], we need to proceed step by step with careful consideration, particularly because the expression involves square roots which can be tricky.
Let's look at the expression [tex]\(\frac{\sqrt{x + a} - \sqrt{3x - 9}}{x - a}\)[/tex]:
1. Direct Substitution: If we directly substitute [tex]\(x = a\)[/tex] into the expression, it will result in [tex]\(\frac{0}{0}\)[/tex], which is an indeterminate form. This suggests that we need to simplify the expression by some other means.
2. Rationalizing the Numerator: To simplify the expression, we will rationalize the numerator. We do this by multiplying the numerator and the denominator by the conjugate of the numerator: [tex]\(\sqrt{x + a} + \sqrt{3x - 9}\)[/tex].
3. Apply the Conjugate: Let's multiply the numerator and denominator by [tex]\(\sqrt{x + a} + \sqrt{3x - 9}\)[/tex]:
[tex]\[
\frac{\left( \sqrt{x + a} - \sqrt{3x - 9} \right) \left( \sqrt{x + a} + \sqrt{3x - 9} \right)}{\left( x - a \right) \left( \sqrt{x + a} + \sqrt{3x - 9} \right)}
\][/tex]
4. Simplify the Numerator: The numerator simplifies as follows:
[tex]\[
\left( \sqrt{x + a} - \sqrt{3x - 9} \right) \left( \sqrt{x + a} + \sqrt{3x - 9} \right) = (x + a) - (3x - 9) = x + a - 3x + 9 = -2x + a + 9
\][/tex]
So the expression becomes:
[tex]\[
\frac{-2x + a + 9}{(x - a) (\sqrt{x + a} + \sqrt{3x - 9})}
\][/tex]
5. Simplify the Entire Expression: Now, we carefully factor and simplify the expression further:
[tex]\[
\frac{-2(x - a + \frac{a + 9}{-2})}{(x - a) (\sqrt{x + a} + \sqrt{3x - 9})}
\][/tex]
This simplification shows that as [tex]\(x\)[/tex] approaches [tex]\(a\)[/tex]:
[tex]\[
\frac{k_1 (x - a)}{(x - a) (\sqrt{x + a} + \sqrt{3x - 9})}
\][/tex]
Thus, simplifying further we will be left with a simpler function. However, we rely now on the structural understanding of limits and what's behind the scenes.
6. Factor and Cancel Terms: When we factor out the common [tex]\(x - a\)[/tex] in the numerator and the denominator, we end up with:
[tex]\[
\frac{-2}{\sqrt{x + a} + \sqrt{3x - 9}} \text{ specifically at the limit point: } \frac{\text{by handling rational properties } \sqrt{x + a} - signs}
\][/tex]
Finally, let’s evaluate this at [tex]\(x \to a\)[/tex]:
Resulting in a generalized resolved form:
[tex]\[
oosign(sqrt(2)sqrt(a) - sqrt(3)*sqrt(a-3))
\][/tex]
This gives us our convergence, where [tex]\(oo\)[/tex] is considering the structure and dividing through a non-trivial zero.
