Find $\lim_{x \to 4} \frac{(1-\sin(\pi x))}{(x-4)^2}$
Step-by-Step Solution
Key Concept: Substitution simplifies logarithmic expressions in limits, combined with L'Hôpital's rule to resolve indeterminate forms.
Let $3^t = u$, so $t = \log_3 u$ and $t \to 2$ means $u \to 9$. We compute $\lim_{u \to 9} \frac{u^2 - 81}{\log_3 u - 2} = \lim_{u \to 9} \frac{(u-9)(u+9)}{\frac{\ln u - \ln 9}{\ln 3}}$. Using L'Hôpital's rule or simplification: $\lim_{u \to 9} \frac{(u-9)(u+9)\ln 3}{\ln u - \ln 9} = \lim_{u \to 9} \frac{(u+9)\ln 3}{\frac{1}{u}} \cdot (u-9) = 18 \cdot \ln 3 / (1/9) \cdot \text{differentiation} = -36$.
Correct Answer: -36