The value of $\int_0^1 \lim_{n \to \infty} \sum_{k=0}^n \frac{x^{k+2^k}}{k!} dx$ is:
Step-by-Step Solution
Key Concept: Functional equations combined with differentiation reveal the exponential nature of $f(x)$.
From the functional equation $f(xy) = f(x)f(y)$, setting $x = y = 1$ gives $f(1) = f(1)^2$, so $f(1) = 1$. Differentiating the functional equation with respect to $x$ yields $yf'(xy) = f'(x)f(y)$. At $x = 1$, this gives $f'(y) = f'(1)f(y)$. Integrating $\frac{f'(x)}{f(x)} = f'(1)$ yields $\ln f(x) = f'(1) \cdot x + C$, which gives $f(x) = e^{f'(1) \cdot x}$.
Correct Answer: 1