\(\displaystyle \forall n\in\mathbb{Z}^+\), let \(\displaystyle f_n\) be the function on the interval from [0,1] by \(\displaystyle f_n(x)=\frac{x^n}{1+x^n}\). Which of the following statements are true?

1. The sequence {\(\displaystyle f_n\)} converges pointwise on [0,1] to a limit function \(\displaystyle f\).

\(\displaystyle \lim_{n\to\infty}\frac{x^n}{1+x^n}=\lim_{n\to\infty}\frac{nx^{n-1}}{nx^{n-1}}=0=f(x)\)

\(\displaystyle \lim_{n\to\infty}\frac{1^n}{1+1^n}=\frac{1}{2}=f(x)\)

True

2. The sequence {\(\displaystyle f_n\)} converges uniformly on [0,1] to a limit function \(\displaystyle f\).

False

3. \(\displaystyle \lim_{n\to\infty}\int_0^1f_n(x)dx=\int_0^1(\lim_{n\to\infty}f_n(x))dx\)

Don't know

1. The sequence {\(\displaystyle f_n\)} converges pointwise on [0,1] to a limit function \(\displaystyle f\).

\(\displaystyle \lim_{n\to\infty}\frac{x^n}{1+x^n}=\lim_{n\to\infty}\frac{nx^{n-1}}{nx^{n-1}}=0=f(x)\)

\(\displaystyle \lim_{n\to\infty}\frac{1^n}{1+1^n}=\frac{1}{2}=f(x)\)

True

2. The sequence {\(\displaystyle f_n\)} converges uniformly on [0,1] to a limit function \(\displaystyle f\).

False

3. \(\displaystyle \lim_{n\to\infty}\int_0^1f_n(x)dx=\int_0^1(\lim_{n\to\infty}f_n(x))dx\)

Don't know

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