Evaluate the following limits :
`lim_(x to 0)(e^(x)-sinx-1)/x`

Answer

To evaluate the limit

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

we can follow these steps:

### Step 1: Rewrite the limit expression
We start by rewriting the limit expression:

\[ \lim_{x \to 0} \frac{e^x - 1 - \sin x}{x}. \]

### Step 2: Split the limit
We can split the limit into two parts:

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

### Step 3: Evaluate the first limit
Using the standard limit result:

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

we find that:

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

### Step 4: Evaluate the second limit
Using another standard limit result:

\[ \lim_{x \to 0} \frac{\sin x}{x} = 1, \]

we find that:

\[ \lim_{x \to 0} \frac{\sin x}{x} = 1. \]

### Step 5: Combine the results
Now we can combine the results from Steps 3 and 4:

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

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

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

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