Evaluate : `lim_(x to 0)(e^(x)-1)/sqrt(1-cosx)`.

Answer

To evaluate the limit \( \lim_{x \to 0} \frac{e^x - 1}{\sqrt{1 - \cos x}} \), we can follow these steps:

### Step 1: Identify the limit form
As \( x \) approaches 0, both the numerator \( e^x - 1 \) and the denominator \( \sqrt{1 - \cos x} \) approach 0. This gives us the indeterminate form \( \frac{0}{0} \).

**Hint:** When you encounter \( \frac{0}{0} \) form, consider applying L'Hôpital's Rule or using known limits.

### Step 2: Use known limits
We know the following limits:
1. \( \lim_{x \to 0} \frac{e^x - 1}{x} = 1 \)
2. \( \lim_{x \to 0} \frac{1 - \cos x}{x^2} = \frac{1}{2} \)

### Step 3: Rewrite the limit
We can rewrite the limit as:
\[ \lim_{x \to 0} \frac{e^x - 1}{\sqrt{1 - \cos x}} = \lim_{x \to 0} \frac{e^x - 1}{x} \cdot \frac{x}{\sqrt{1 - \cos x}} \]

### Step 4: Substitute known limits
Now, we can substitute the known limits:
- From the first limit, as \( x \to 0 \), \( \frac{e^x - 1}{x} \) approaches 1.
- For \( \sqrt{1 - \cos x} \), we can express it using the second limit:
\[ \sqrt{1 - \cos x} = \sqrt{x^2 \cdot \frac{1 - \cos x}{x^2}} \approx \sqrt{x^2 \cdot \frac{1}{2}} = \frac{x}{\sqrt{2}} \text{ as } x \to 0 \]

### Step 5: Substitute back into the limit
Now, substituting back into our limit:
\[ \lim_{x \to 0} \frac{e^x - 1}{\sqrt{1 - \cos x}} = \lim_{x \to 0} \frac{e^x - 1}{x} \cdot \frac{x}{\sqrt{1 - \cos x}} = 1 \cdot \frac{x}{\frac{x}{\sqrt{2}}} = \sqrt{2} \]

### Final Answer
Thus, the limit evaluates to:
\[ \lim_{x \to 0} \frac{e^x - 1}{\sqrt{1 - \cos x}} = \sqrt{2} \]

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