Exercise 7.7

Question

Exercise 7: 
    For $n=1,2,3,\ldots$, $x$ real, put $$f_n(x)=\frac{x}{1+nx^2}.$$ Show that $\{f_n\}$ converges uniformly to a function $f$, and that the equation $$f'(x)=\lim_{nightarrow\infty}f_n'(x)$$
    is correct if $x
eq 0$, but false if $x=0$.

analambanomenos · Accepted Answer

Calculating the minimum and maximum values of $f_n$ using elementary calculus, we get that $$\big|f_n(x)\big|\le\frac{1}{2\sqrt{n}},$$ so that $f_n(x)$ converges uniformly to the constant
    function $f(x)=0$ on $\mathbf R$. For $x
e0$, $$f_n'(x)=\frac{1-nx^2}{(1+nx^2)^2}$$ converges to $f'(x)=0$, but $f_n'(0)=1$ does not.

Exercise 7.7

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Exercise 7.7

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