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Consider the following statements: Statement 1: If \(\lim_{x\to\alpha}\frac{\sin(f(x))}{x-\alpha}\), where \(f(x)=ax^2+bx+c\), exists as a finite nonzero number, then \(\lim_{x\to\alpha}\frac{1/f(x)-1}{1/f(x)+1}\) does not exist. Statement 2: The limit \(\lim_{x\to\alpha}\frac{\sin(f(x))}{x-\alpha}\) can be finite only in the indeterminate form \(0/0\).

1. Statement 1 is correct, Statement 2 is correct, and Statement 2 correctly explains Statement 1
2. Statement 1 is correct, Statement 2 is correct, but Statement 2 does not correctly explain Statement 1
3. Statement 1 is incorrect, Statement 2 is correct
4. Statement 1 is correct, Statement 2 is incorrect

Correct answer: Statement 1 is correct, Statement 2 is correct, and Statement 2 correctly explains Statement 1

Solution

Statement 1 is correct because the limit involving \\( rac{ ext{sin}(f(x))}{x- ext{alpha}}\\) implies that \\( ext{sin}(f(x))\\) approaches zero as \\( ext{x}\\) approaches \\( ext{alpha}\\), leading to a finite nonzero limit. Statement 2 is also correct as the limit can only be finite in the indeterminate form \\( rac{0}{0}\\), which aligns with the conclusion of Statement 1.

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