If `f(x) = (e^([x] + |x|) -3)/([x] + |x|+ 1)` , then: (where [.] represents greatest integer function)

Options

A. `lim_(x to 0^(+)) f(x) = -2`
B. `lim_(x to 0^(+)) f(x) = 0`
C. `lim_(x to 0^(+)) f(x) = 2`
D. `lim_(x to 0^(+)) f(x)` exist

Explanation

To solve the function \( f(x) = \frac{e^{[x] + |x|} - 3}{[x] + |x| + 1} \), where \([x]\) represents the greatest integer function, we will analyze the function step by step.

### Step 1: Understand the components of the function
The function consists of two main components: the greatest integer function \([x]\) and the absolute value function \(|x|\).

### Step 2: Analyze the case for \( x \geq 0 \)
For \( x \geq 0 \):
- The greatest integer function \([x] = n\) where \( n \) is the largest integer less than or equal to \( x \).
- The absolute value \(|x| = x\).

Thus, for \( x \) in the interval \([n, n+1)\):
- \([x] = n\)
- \(|x| = x\)

So, we can rewrite the function as:
\[ f(x) = \frac{e^{n + x} - 3}{n + x + 1} \]

### Step 3: Evaluate the limit as \( x \) approaches 0 from the right
Now, we will evaluate the limit as \( x \) approaches \( 0^+ \):
- Here, \([x] = 0\) and \(|x| = x\).

Substituting these values into the function:
\[ f(x) = \frac{e^{0 + x} - 3}{0 + x + 1} = \frac{e^{x} - 3}{x + 1} \]

### Step 4: Calculate the limit
Now we need to calculate:
\[ \lim_{x \to 0^+} f(x) = \lim_{x \to 0^+} \frac{e^{x} - 3}{x + 1} \]

As \( x \to 0^+ \):
- \( e^{x} \to e^0 = 1 \)
- Thus, the limit becomes:
\[ \lim_{x \to 0^+} \frac{1 - 3}{0 + 1} = \frac{-2}{1} = -2 \]

### Conclusion
Therefore, the value of \( f(x) \) as \( x \) approaches \( 0^+ \) is:
\[ \lim_{x \to 0^+} f(x) = -2 \]