`lim_(xrarr0) (sin4x)/(1-sqrt(1-x))`, is

Options

A. 4
B. 8
C. 10
D. 2

Answer

To solve the limit \( \lim_{x \to 0} \frac{\sin 4x}{1 - \sqrt{1 - x}} \), we will follow these steps:

### Step 1: Rationalize the Denominator
We start by rationalizing the denominator \( 1 - \sqrt{1 - x} \). The rationalizing factor is \( 1 + \sqrt{1 - x} \). Thus, we multiply the numerator and the denominator by this factor:

\[ \lim_{x \to 0} \frac{\sin 4x}{1 - \sqrt{1 - x}} \cdot \frac{1 + \sqrt{1 - x}}{1 + \sqrt{1 - x}} = \lim_{x \to 0} \frac{\sin 4x (1 + \sqrt{1 - x})}{(1 - \sqrt{1 - x})(1 + \sqrt{1 - x})} \]

### Step 2: Simplify the Denominator
Using the difference of squares, we simplify the denominator:

\[ (1 - \sqrt{1 - x})(1 + \sqrt{1 - x}) = 1^2 - (\sqrt{1 - x})^2 = 1 - (1 - x) = x \]

So, we can rewrite the limit as:

\[ \lim_{x \to 0} \frac{\sin 4x (1 + \sqrt{1 - x})}{x} \]

### Step 3: Split the Limit
Now we can split the limit into two parts:

\[ \lim_{x \to 0} \frac{\sin 4x}{x} \cdot \lim_{x \to 0} (1 + \sqrt{1 - x}) \]

### Step 4: Evaluate the First Limit
We know from the standard limit that:

\[ \lim_{x \to 0} \frac{\sin kx}{x} = k \]

In our case, \( k = 4 \):

\[ \lim_{x \to 0} \frac{\sin 4x}{x} = 4 \]

### Step 5: Evaluate the Second Limit
Next, we evaluate the second limit:

\[ \lim_{x \to 0} (1 + \sqrt{1 - x}) = 1 + \sqrt{1 - 0} = 1 + 1 = 2 \]

### Step 6: Combine the Results
Now we can combine the results from the two limits:

\[ 4 \cdot 2 = 8 \]

### Final Answer
Thus, the limit is:

\[ \lim_{x \to 0} \frac{\sin 4x}{1 - \sqrt{1 - x}} = 8 \]