Let $f(x) = \dfrac{\sin(\pi x^4) + (x+2)^n \tan(\pi x)/(x+1)}{1 + (x+2)^n - x^4}$. Find $\displaystyle\lim_{x \to -1} f(x)$ (as $n \to \infty$).
Step-by-Step Solution
Key Concept: Near $x=-1$, $(x+2)^n\to 1^n=1$ but examine what happens as $n\to\infty$; dominant term analysis.
As $x\to -1$: $x+2\to 1$, so $(x+2)^n\to 1$ for any finite $n$. Then $f(x)\to \frac{\sin\pi+1\cdot\tan(-\pi)/0 + ...}{1+1-1}$. More carefully: $\sin(\pi\cdot 1) = 0$ and $\tan(\pi x)/(x+1)\to\pi$ as $x\to -1$. So $\lim = \frac{0+1\cdot\pi}{1} = \pi$.
Correct Answer: A