`lim_(xtoa^(-))(sqrt(x-b)-sqrt(a-b))/((x^(2)-a^(2))),(agtb)` is

Options

A. `1/(4a)`
B. `1/(asqrt(a-b))`
C. `1/(2asqrt(a-b))`
D. `1/(4asqrt(a-b))`

Answer

To solve the limit \( \lim_{x \to a^-} \frac{\sqrt{x-b} - \sqrt{a-b}}{x^2 - a^2} \), we can follow these steps:

### Step 1: Substitute \( x = a \)
First, we substitute \( x = a \) into the limit expression to check if it results in an indeterminate form.

\[ \sqrt{a-b} - \sqrt{a-b} = 0 \]
\[ a^2 - a^2 = 0 \]

This gives us the form \( \frac{0}{0} \), which is indeterminate.

### Step 2: Apply L'Hôpital's Rule
Since we have an indeterminate form, we can apply L'Hôpital's Rule, which states that we can take the derivative of the numerator and the denominator.

#### Differentiate the Numerator:
The numerator is \( \sqrt{x-b} - \sqrt{a-b} \). The derivative of \( \sqrt{x-b} \) is:
\[ \frac{1}{2\sqrt{x-b}} \]
The derivative of \( \sqrt{a-b} \) is 0 (since it is a constant).

Thus, the derivative of the numerator is:
\[ \frac{1}{2\sqrt{x-b}} \]

#### Differentiate the Denominator:
The denominator is \( x^2 - a^2 \). The derivative is:
\[ 2x \]

### Step 3: Rewrite the Limit
Now we can rewrite the limit using the derivatives we found:
\[ \lim_{x \to a^-} \frac{\frac{1}{2\sqrt{x-b}}}{2x} \]

### Step 4: Simplify the Expression
This simplifies to:
\[ \lim_{x \to a^-} \frac{1}{4x\sqrt{x-b}} \]

### Step 5: Substitute \( x = a \) Again
Now we substitute \( x = a \):
\[ \frac{1}{4a\sqrt{a-b}} \]

### Final Answer
Thus, the limit is:
\[ \frac{1}{4a\sqrt{a-b}} \]