1. **State the problem:** Find the limit $$\lim_{x \to 0} \frac{e^{2x} - 1}{x}$$. 2. **Recall the formula and rules:** The limit resembles the derivative definition of the exponential function at 0. We use the fact that $$\lim_{x \to 0} \frac{e^{kx} - 1}{x} = k$$ for any constant $k$. 3. **Apply the limit:** Here, $k=2$, so $$\lim_{x \to 0} \frac{e^{2x} - 1}{x} = 2$$. 4. **Explain:** This is because the derivative of $e^{2x}$ at $x=0$ is $2e^{2 \cdot 0} = 2 \cdot 1 = 2$. **Final answer:** $$2$$