Evaluate the limits, if exist`lim_(x rarr 0)(x(e^x-1)/(1-cosx))`

Explanation

To evaluate the limit

\[ \lim_{x \to 0} \frac{x(e^x - 1)}{1 - \cos x}, \]

we will follow these steps:

### Step 1: Rewrite the denominator using a trigonometric identity
We can use the identity for cosine:

\[ 1 - \cos x = 2 \sin^2\left(\frac{x}{2}\right). \]

Thus, we can rewrite the limit as:

\[ \lim_{x \to 0} \frac{x(e^x - 1)}{2 \sin^2\left(\frac{x}{2}\right)}. \]

### Step 2: Factor out constants
We can factor out the constant \( \frac{1}{2} \) from the limit:

\[ \lim_{x \to 0} \frac{x(e^x - 1)}{2 \sin^2\left(\frac{x}{2}\right)} = \frac{1}{2} \lim_{x \to 0} \frac{x(e^x - 1)}{\sin^2\left(\frac{x}{2}\right)}. \]

### Step 3: Use standard limits
We know from standard limits that:

\[ \lim_{x \to 0} \frac{e^x - 1}{x} = 1, \]

and

\[ \lim_{x \to 0} \frac{\sin\left(\frac{x}{2}\right)}{\frac{x}{2}} = 1 \implies \lim_{x \to 0} \frac{\sin^2\left(\frac{x}{2}\right)}{\left(\frac{x}{2}\right)^2} = 1. \]

### Step 4: Rewrite the limit
Now we can rewrite our limit using these standard results. We will multiply and divide by \( x^2 \):

\[ \lim_{x \to 0} \frac{e^x - 1}{x} \cdot \frac{x^2}{\sin^2\left(\frac{x}{2}\right)}. \]

### Step 5: Substitute the limits
Substituting the limits we have:

\[ \frac{1}{2} \cdot 1 \cdot \lim_{x \to 0} \frac{x^2}{\sin^2\left(\frac{x}{2}\right)} = \frac{1}{2} \cdot 1 \cdot 4 = 2. \]

### Final Answer
Thus, the limit evaluates to:

\[ \lim_{x \to 0} \frac{x(e^x - 1)}{1 - \cos x} = 2. \]

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