Question

Suppose that for n ∈ N, f_{n}(x)={-1 , -1le x le -1/n
{nx , |x| le 1/n,
{1 , 1/n le x le 1
a)Find the limit function for {f_{n}}
b)Determine whether {f_{n}} converges uniformly on [-1,1]
c)Show that int_{-1}^{1} lim_{n rightarrow infty} f_{n}(x)dx=lim_{n rightarrow infty } int_{-1}^{1} f_{n}(x)dx

Answer 1

c) \[ \int_{-1}^{1} \lim_{n \rightarrow \infty} f_{n}(x)dx=\lim_{n \rightarrow \infty } \int_{-1}^{1} f_{n}(x)dx\]

Answer 2

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