Show that the limit and derivative cannot be interchanged when f_n(x) = (sin (nx)) / n

by RoRi

March 31, 2016

For each positive integer $n$ and and all real $x$ define

$f_n (x) = \frac{\sin (nx)}{n}, \qquad f(x) = \lim_{n \to \infty} f_n(x).$

Prove that

$\lim_{n \to \infty} f_n'(0) \neq f'(0).$

Proof. First, we have

$f_n (x) = \frac{\sin (nx)}{n} \quad \implies \quad f_n'(x) = \cos (nx) \quad \implies \quad f_n'(0) = 1$

for all $n$ . Hence,

$\lim_{n \to \infty} f_n'(0) = 1.$

On the other hand,

$f(x) = \lim_{n \to \infty} f_n(x) = \lim_{n \to \infty} \frac{\sin (nx)}{n} = 0 \implies f'(0) = 0.$

Hence,

$\lim_{n \to \infty} f_n'(0) \neq f'(0). \qquad \blacksquare$

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