`lim _( x to 0) (e ^(sin (x ))-1)/(x )` equals to :

Options

A. 1
B. 2
C. e
D. does not exist

Answer

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

### Step 1: Recognize the Limit Form
We start with the limit:
\[ L = \lim_{x \to 0} \frac{e^{\sin x} - 1}{x} \]
As \( x \) approaches 0, both the numerator and denominator approach 0, creating a \( \frac{0}{0} \) indeterminate form.

### Step 2: Use the Sine Function
To resolve the indeterminate form, we can multiply and divide by \( \sin x \):
\[ L = \lim_{x \to 0} \frac{e^{\sin x} - 1}{\sin x} \cdot \frac{\sin x}{x} \]

### Step 3: Split the Limit
Now we can separate the limit into two parts:
\[ L = \lim_{x \to 0} \frac{e^{\sin x} - 1}{\sin x} \cdot \lim_{x \to 0} \frac{\sin x}{x} \]

### Step 4: Evaluate the Second Limit
We know from standard limits that:
\[ \lim_{x \to 0} \frac{\sin x}{x} = 1 \]

### Step 5: Evaluate the First Limit
Now we need to evaluate:
\[ \lim_{x \to 0} \frac{e^{\sin x} - 1}{\sin x} \]
Let \( y = \sin x \). As \( x \to 0 \), \( y \to 0 \) as well. Thus, we can rewrite the limit:
\[ \lim_{y \to 0} \frac{e^y - 1}{y} \]

### Step 6: Apply the Exponential Limit
We know from calculus that:
\[ \lim_{y \to 0} \frac{e^y - 1}{y} = 1 \]

### Step 7: Combine the Results
Now we can combine our results:
\[ L = 1 \cdot 1 = 1 \]

### Final Answer
Thus, the limit is:
\[ \lim_{x \to 0} \frac{e^{\sin x} - 1}{x} = 1 \]