`lim_(xrarr1)(sqrt(1+x)-sqrt(1-x))/(1+x)` is equal to

Options

A. (i) `(1)/(sqrt(3))`
B. (ii) `(1)/(sqrt(2))`
C. (iii) `sqrt(3)`
D. (iv) `sqrt(2)`

Answer

To solve the limit \( \lim_{x \to 1} \frac{\sqrt{1+x} - \sqrt{1-x}}{1+x} \), we can follow these steps:

### Step 1: Identify the limit
We start with the limit expression:
\[ \lim_{x \to 1} \frac{\sqrt{1+x} - \sqrt{1-x}}{1+x} \]

### Step 2: Substitute \( x = 1 \)
If we directly substitute \( x = 1 \), we get:
\[ \frac{\sqrt{1+1} - \sqrt{1-1}}{1+1} = \frac{\sqrt{2} - 0}{2} = \frac{\sqrt{2}}{2} \]
However, we need to check if this leads to an indeterminate form.

### Step 3: Rationalize the numerator
To simplify the expression, we can rationalize the numerator:
\[ \frac{\sqrt{1+x} - \sqrt{1-x}}{1+x} \cdot \frac{\sqrt{1+x} + \sqrt{1-x}}{\sqrt{1+x} + \sqrt{1-x}} = \frac{(1+x) - (1-x)}{(1+x)(\sqrt{1+x} + \sqrt{1-x})} \]
This simplifies to:
\[ \frac{2x}{(1+x)(\sqrt{1+x} + \sqrt{1-x})} \]

### Step 4: Substitute \( x = 1 \) again
Now we can substitute \( x = 1 \) into the simplified expression:
\[ \lim_{x \to 1} \frac{2x}{(1+x)(\sqrt{1+x} + \sqrt{1-x})} = \frac{2 \cdot 1}{(1+1)(\sqrt{1+1} + \sqrt{1-1})} = \frac{2}{2(\sqrt{2} + 0)} = \frac{2}{2\sqrt{2}} = \frac{1}{\sqrt{2}} \]

### Conclusion
Thus, the limit evaluates to:
\[ \lim_{x \to 1} \frac{\sqrt{1+x} - \sqrt{1-x}}{1+x} = \frac{1}{\sqrt{2}} \]

### Final Answer
The final answer is:
\[ \frac{1}{\sqrt{2}} \]