1. **State the problem:** Find the limit $$\lim_{x \to 0} \frac{e^2 - 1}{x}$$. 2. **Analyze the expression:** The numerator $$e^2 - 1$$ is a constant (approximately 6.389056), and the denominator approaches 0 as $$x \to 0$$. 3. **Evaluate the limit:** Since the numerator is a nonzero constant and the denominator approaches 0, the fraction grows without bound. 4. **Determine the sign:** As $$x \to 0^+$$, the denominator is positive and the fraction tends to $$+\infty$$. As $$x \to 0^-$$, the denominator is negative and the fraction tends to $$-\infty$$. 5. **Conclusion:** The limit does not exist because the left-hand and right-hand limits are not equal. **Final answer:** $$\lim_{x \to 0} \frac{e^2 - 1}{x}$$ does not exist.