Let $T = \lim_{x \to 3} \left( \frac{f(x+2)}{f(3)} \right)^{\frac{18}{(x-3)^2}}$ ; $1^\infty$ form

$\Rightarrow T = e^{\lim \frac{18}{(x-3)^2} \cdot \frac{f(x+2) - f(3)}{f(3)}}$

$\Rightarrow T = e^{\lim \frac{18}{(x-3)^2} \cdot \frac{f(x+2) - f(3)}{18}}$

$\Rightarrow T = e^{\lim \frac{f(x+2) - f(3)}{(x-3)^2}}$ ; $0/0$ form apply L’Hospital

$\Rightarrow T = e^{\lim \frac{f'(x+2)}{2(x-3)}}$ ; $0/0$ form apply L’Hospital

$\Rightarrow T = e^{\lim \frac{f''(x+2)}{2}} = e^2$

$\Rightarrow \log_e (T) = 2$