Let `f(x)=(sqrt(1+sinx)-sqrt(1-sinx))/(tanx) , x ne 0` Then `lim_(x to 0) f(x)` is equal to

Answer

To find the limit of the function \( f(x) = \frac{\sqrt{1 + \sin x} - \sqrt{1 - \sin x}}{\tan x} \) as \( x \) approaches 0, we can follow these steps:

### Step 1: Substitute \( x = 0 \)
First, we substitute \( x = 0 \) into the function to check if we get an indeterminate form:

\[ f(0) = \frac{\sqrt{1 + \sin(0)} - \sqrt{1 - \sin(0)}}{\tan(0)} = \frac{\sqrt{1 + 0} - \sqrt{1 - 0}}{0} = \frac{1 - 1}{0} = \frac{0}{0} \]

This is an indeterminate form \( \frac{0}{0} \).

### Step 2: Apply L'Hôpital's Rule
Since we have an indeterminate form, we can apply L'Hôpital's Rule, which states that if \( \lim_{x \to c} \frac{f(x)}{g(x)} = \frac{0}{0} \) or \( \frac{\infty}{\infty} \), then:

\[ \lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)} \]

Here, \( f(x) = \sqrt{1 + \sin x} - \sqrt{1 - \sin x} \) and \( g(x) = \tan x \).

### Step 3: Differentiate the Numerator and Denominator
Now we differentiate the numerator and denominator separately.

**Numerator:**
Using the chain rule:

\[ f'(x) = \frac{1}{2\sqrt{1 + \sin x}} \cdot \cos x - \frac{1}{2\sqrt{1 - \sin x}} \cdot \cos x \]

This simplifies to:

\[ f'(x) = \cos x \left( \frac{1}{2\sqrt{1 + \sin x}} + \frac{1}{2\sqrt{1 - \sin x}} \right) \]

**Denominator:**
The derivative of \( g(x) = \tan x \) is:

\[ g'(x) = \sec^2 x \]

### Step 4: Rewrite the Limit
Now we can rewrite the limit using L'Hôpital's Rule:

\[ \lim_{x \to 0} f(x) = \lim_{x \to 0} \frac{f'(x)}{g'(x)} = \lim_{x \to 0} \frac{\cos x \left( \frac{1}{2\sqrt{1 + \sin x}} + \frac{1}{2\sqrt{1 - \sin x}} \right)}{\sec^2 x} \]

### Step 5: Simplify the Expression
We can simplify this limit:

\[ = \lim_{x \to 0} \frac{\cos^3 x}{2\sqrt{1 + \sin x} + 2\sqrt{1 - \sin x}} \]

### Step 6: Substitute \( x = 0 \) Again
Now we substitute \( x = 0 \):

\[ = \frac{1^3}{2\sqrt{1 + 0} + 2\sqrt{1 - 0}} = \frac{1}{2(1) + 2(1)} = \frac{1}{4} \]

### Final Result
Thus, the limit is:

\[ \lim_{x \to 0} f(x) = \frac{1}{4} \]