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Let \[f(x)=\lim_{n\to\infty}\frac{\tan(\pi x^2)+(x+1)^n\sin x}{x^2+(x+1)^n}.\] Then which statement is true about the function at \(x=0\)?

1. The function is continuous at \(x=0\).
2. The function is differentiable at \(x=0\).
3. The function is continuous at \(x=0\) but not differentiable there.
4. None of the above.

Correct answer: The function is continuous at \(x=0\).

Solution

The function is continuous at x=0 because as n approaches infinity, the term (x+1)^n dominates both the numerator and denominator, leading to a limit that exists and equals a finite value when evaluated at x=0.

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