Evaluate the limits, if exist `lim_(x rarr 0)(e^(2+x)-e^2)/x`.

Explanation

To evaluate the limit \( \lim_{x \to 0} \frac{e^{2+x} - e^2}{x} \), we can follow these steps:

### Step 1: Rewrite the expression
We start with the limit expression:
\[ \lim_{x \to 0} \frac{e^{2+x} - e^2}{x} \]
We can factor out \( e^2 \) from the numerator:
\[ = \lim_{x \to 0} \frac{e^2(e^x - 1)}{x} \]

### Step 2: Simplify the limit
Now, we can rewrite the limit as:
\[ = e^2 \lim_{x \to 0} \frac{e^x - 1}{x} \]

### Step 3: Evaluate the limit
The limit \( \lim_{x \to 0} \frac{e^x - 1}{x} \) is a standard limit that equals 1. Therefore, we have:
\[ = e^2 \cdot 1 = e^2 \]

### Conclusion
Thus, the limit is:
\[ \lim_{x \to 0} \frac{e^{2+x} - e^2}{x} = e^2 \]

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