MATCH THE FOLLOWING:
(A) $\lim_{n \to \infty} \int_0^1 \frac{2}{n+1}\left[1+\frac{t}{n+1}\right]^n dt$
Step-by-Step Solution
Key Concept: Use the two differential equation conditions to form a system: one gives the sum and one gives the difference of the functions.
Given $f'(x) + g'(x) = 0$, we have $f(x) + g(x) = k$. Substituting $x = 0$ gives $f(0) + g(0) = 6$, so $k = 6$. From $f'(x) - g'(x) = 2(f(x) - g(x))$, we get $\frac{f'(x) - g'(x)}{f(x) - g(x)} = 2$. Integrating yields $\ln|f(x) - g(x)| = 2x + c$. Using $x = 0$: $c = \ln 4$, so $f(x) - g(x) = 4e^{2x}$. Solving the system: $f(x) = 3 + 2e^{2x}$ and $g(x) = 3 - 2e^{2x}$. Therefore, $f(\ln 2) + g(\ln 3) = 11 + (-15) = 4$.
Correct Answer: [A-r] [B-p] [C-s] [D-q]