If f(4)= 4, f'(4) =1 then `lim_(x to 4) 2((2-sqrtf(x))/ (2 - sqrtx))` is equal to

Options

A. 0
B. 2
C. `-1`
D. 2

Answer

To solve the limit \( \lim_{x \to 4} 2 \frac{2 - \sqrt{f(x)}}{2 - \sqrt{x}} \) given that \( f(4) = 4 \) and \( f'(4) = 1 \), we can follow these steps:

### Step 1: Substitute the limit
First, we substitute \( x = 4 \) into the limit expression:

\[ \lim_{x \to 4} 2 \frac{2 - \sqrt{f(x)}}{2 - \sqrt{x}} = 2 \frac{2 - \sqrt{f(4)}}{2 - \sqrt{4}} \]

### Step 2: Evaluate \( f(4) \) and \( \sqrt{4} \)
From the problem, we know:

- \( f(4) = 4 \)
- \( \sqrt{4} = 2 \)

Substituting these values into the expression gives:

\[ = 2 \frac{2 - \sqrt{4}}{2 - 2} = 2 \frac{2 - 2}{2 - 2} = 2 \frac{0}{0} \]

### Step 3: Identify the indeterminate form
We see that we have an indeterminate form \( \frac{0}{0} \). Therefore, we can apply L'Hôpital's Rule.

### Step 4: Apply L'Hôpital's Rule
According to L'Hôpital's Rule, we differentiate the numerator and denominator:

\[ \lim_{x \to 4} 2 \frac{2 - \sqrt{f(x)}}{2 - \sqrt{x}} = 2 \lim_{x \to 4} \frac{-\frac{1}{2\sqrt{f(x)}} f'(x)}{-\frac{1}{2\sqrt{x}}} \]

### Step 5: Simplify the expression
This simplifies to:

\[ = 2 \lim_{x \to 4} \frac{f'(x)}{\sqrt{f(x)} \cdot \sqrt{x}} \]

### Step 6: Substitute \( x = 4 \) again
Now we substitute \( x = 4 \):

\[ = 2 \cdot \frac{f'(4)}{\sqrt{f(4)} \cdot \sqrt{4}} = 2 \cdot \frac{f'(4)}{\sqrt{4} \cdot 2} \]

### Step 7: Use given values
We know:

- \( f'(4) = 1 \)
- \( \sqrt{4} = 2 \)

Substituting these values gives:

\[ = 2 \cdot \frac{1}{2 \cdot 2} = 2 \cdot \frac{1}{4} = \frac{2}{4} = \frac{1}{2} \]

### Final Answer
Thus, the limit is:

\[ \lim_{x \to 4} 2 \frac{2 - \sqrt{f(x)}}{2 - \sqrt{x}} = 2 \]