Evaluate `lim_(xtoa) (sqrt(3x-a)-sqrt(x+a))/(x-a).`

Explanation

To evaluate the limit

\[ \lim_{x \to a} \frac{\sqrt{3x - a} - \sqrt{x + a}}{x - a}, \]

we start by substituting \(x = a\) into the expression. This gives us:

\[ \frac{\sqrt{3a - a} - \sqrt{a + a}}{a - a} = \frac{\sqrt{2a} - \sqrt{2a}}{0} = \frac{0}{0}. \]

Since we have an indeterminate form \(0/0\), we will use the rationalization technique to simplify the expression.

### Step 1: Rationalize the numerator

We multiply the numerator and denominator by the conjugate of the numerator:

\[ \lim_{x \to a} \frac{(\sqrt{3x - a} - \sqrt{x + a})(\sqrt{3x - a} + \sqrt{x + a})}{(x - a)(\sqrt{3x - a} + \sqrt{x + a})}. \]

This simplifies to:

\[ \lim_{x \to a} \frac{(3x - a) - (x + a)}{(x - a)(\sqrt{3x - a} + \sqrt{x + a})}. \]

### Step 2: Simplify the numerator

Now, simplify the numerator:

\[ 3x - a - x - a = 2x - 2a = 2(x - a). \]

So, we have:

\[ \lim_{x \to a} \frac{2(x - a)}{(x - a)(\sqrt{3x - a} + \sqrt{x + a})}. \]

### Step 3: Cancel common factors

We can cancel \(x - a\) from the numerator and denominator (as long as \(x \neq a\)):

\[ \lim_{x \to a} \frac{2}{\sqrt{3x - a} + \sqrt{x + a}}. \]

### Step 4: Substitute \(x = a\)

Now, we substitute \(x = a\):

\[ \frac{2}{\sqrt{3a - a} + \sqrt{a + a}} = \frac{2}{\sqrt{2a} + \sqrt{2a}} = \frac{2}{2\sqrt{2a}} = \frac{1}{\sqrt{2a}}. \]

### Final Answer

Thus, the limit evaluates to:

\[ \lim_{x \to a} \frac{\sqrt{3x - a} - \sqrt{x + a}}{x - a} = \frac{1}{\sqrt{2a}}. \]

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