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The given mathematical expression is the limit:

\[
\lim_{{x \to a}} \frac{\sqrt{x} - \sqrt{a}}{x - a}
\]

To solve this limit, let's first recognize that directly substituting \( x = a \) leads to an indeterminate form \(\frac{0}{0}\). Therefore, we can apply algebraic manipulation such as multiplying by the conjugate of the numerator to simplify the expression.

1. Multiply the numerator and denominator by the conjugate of the numerator, \(\sqrt{x} + \sqrt{a}\):

\[
\frac{\sqrt{x} - \sqrt{a}}{x - a} \cdot \frac{\sqrt{x} + \sqrt{a}}{\sqrt{x} + \sqrt{a}} = \frac{(\sqrt{x})^2 - (\sqrt{a})^2}{(x - a)(\sqrt{x} + \sqrt{a})}
\]

2. Simplify the numerator using the difference of squares formula:

\[
= \frac{x - a}{(x - a)(\sqrt{x} + \sqrt{a})}
\]

3. Cancel out the common factor \( x - a \) from the numerator and denominator:

\[
= \frac{1}{\sqrt{x} + \sqrt{a}}
\]

4. Now, apply the limit \( x \to a \):

\[
\lim_{{x \to a}} \frac{1}{\sqrt{x} + \sqrt{a}} = \frac{1}{\sqrt{a} + \sqrt{a}} = \frac{1}{2\sqrt{a}}
\]

Thus, the value of the limit is:

\[
\boxed{\frac{1}{2\sqrt{a}}}
\]

Let me know if you would like more details or have any questions!

Here are five related questions you can explore:
1. How can you use L'Hopital's Rule to solve this limit?
2. What is the conjugate and why does multiplying by it simplify this expression?
3. Can you generalize this technique for limits involving square roots?
4. How does this approach compare to other limit-solving techniques?
5. How can this limit help in understanding the derivative of \(\sqrt{x}\)?

**Tip:** Multiplying by the conjugate is a powerful technique when dealing with square roots in limits and simplifies many indeterminate forms.

Evaluate the limit as x approaches a for the expression: (√x - √a) / (x - a)

This detailed explanation shows how to evaluate the limit of (√x - √a) / (x - a) as x approaches a. Learn how algebraic manipulation using the conjugate simplifies indeterminate forms to reach the solution of 1/(2√a). Suitable for students in Grades 10-12.

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