Let `f(x)=(1)/(sqrt(18-x^(2)))`
What is the value of `lim_(xto3) (\f(x)-f(3))/(x-3)`?

Options

A. 0
B. `-(1)/(9)`
C. `(1)/(3)`
D. `(1)/(9)`

Explanation

To solve the limit problem, we will follow these steps:

### Step 1: Identify the function and the limit
We have the function:
\[ f(x) = \frac{1}{\sqrt{18 - x^2}} \]
We need to find:
\[ \lim_{x \to 3} \frac{f(x) - f(3)}{x - 3} \]

### Step 2: Calculate \( f(3) \)
First, we calculate \( f(3) \):
\[ f(3) = \frac{1}{\sqrt{18 - 3^2}} = \frac{1}{\sqrt{18 - 9}} = \frac{1}{\sqrt{9}} = \frac{1}{3} \]

### Step 3: Rewrite the limit expression
Now, substituting \( f(3) \) into our limit expression:
\[ \lim_{x \to 3} \frac{f(x) - \frac{1}{3}}{x - 3} = \lim_{x \to 3} \frac{\frac{1}{\sqrt{18 - x^2}} - \frac{1}{3}}{x - 3} \]

### Step 4: Combine the fractions in the numerator
To combine the fractions in the numerator, we find a common denominator:
\[ \frac{1}{\sqrt{18 - x^2}} - \frac{1}{3} = \frac{3 - \sqrt{18 - x^2}}{3\sqrt{18 - x^2}} \]
Thus, our limit becomes:
\[ \lim_{x \to 3} \frac{\frac{3 - \sqrt{18 - x^2}}{3\sqrt{18 - x^2}}}{x - 3} = \lim_{x \to 3} \frac{3 - \sqrt{18 - x^2}}{3\sqrt{18 - x^2}(x - 3)} \]

### Step 5: Apply L'Hôpital's Rule
As \( x \to 3 \), both the numerator and denominator approach 0, so we can apply L'Hôpital's Rule:
\[ \lim_{x \to 3} \frac{3 - \sqrt{18 - x^2}}{3\sqrt{18 - x^2}(x - 3)} \rightarrow \text{Differentiate the numerator and denominator} \]
The derivative of the numerator \( 3 - \sqrt{18 - x^2} \) is:
\[ \frac{d}{dx}(3 - \sqrt{18 - x^2}) = \frac{x}{\sqrt{18 - x^2}} \]
The derivative of the denominator \( 3\sqrt{18 - x^2}(x - 3) \) using the product rule is:
\[ 3\left(\frac{x}{\sqrt{18 - x^2}}(x - 3) + \sqrt{18 - x^2}\right) \]

### Step 6: Evaluate the limit again
Now substituting \( x = 3 \) into the derivatives:
\[ \text{Numerator: } \frac{3}{\sqrt{9}} = 1 \]
\[ \text{Denominator: } 3\left(\frac{3}{3}(0) + 3\right) = 9 \]
Thus, the limit evaluates to:
\[ \lim_{x \to 3} \frac{1}{9} = \frac{1}{9} \]

### Final Answer
The value of the limit is:
\[ \frac{1}{9} \]