For each positive integer and and all real define

Prove that

*Proof.* First, we have

for all . Hence,

On the other hand,

Hence,

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Stumbling Robot

A Fraction of a Dot
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Show that the limit and derivative cannot be interchanged when *f*_{n}(x) = (sin (nx)) / n

For each positive integer and and all real define

Prove that

*Proof.* First, we have

for all . Hence,

On the other hand,

Hence,