# derivative of limit function diverges from limit of derivatives

For a function sequence, one cannot always change the of http://planetmath.org/node/6209taking limit and differentiating (http://planetmath.org/Differentiate), i.e. it may well be

 $\lim_{n\to\infty}\frac{d}{dx}f_{n}(x)\;\neq\;\frac{d}{dx}\lim_{n\to\infty}f_{n% }(x),$

in the case that a sequence of continuous (and differentiable) functions converges uniformly; cf. Theorem 2 of the parent entry (http://planetmath.org/LimitFunctionOfSequence).

Example.  The function sequence

 $\displaystyle f_{n}(x)\;:=\;\sum_{j=1}^{n}\frac{x^{3}}{(1\!+\!x^{2})^{j}}\;=\;% x-\frac{x}{(1\!+\!x^{2})^{n}}\qquad(n\;=\;1,\,2,\,3,\,\ldots)$ (1)

provides an instance; we consider it on the interval$[-1,\,1]$.  It’s a question of partial sum the converging geometric series

 $\frac{x^{3}}{1\!+\!x^{2}}+\frac{x^{3}}{(1\!+\!x^{2})^{2}}+\frac{x^{3}}{(1\!+\!% x^{2})^{2}}+\ldots$

(although one cannot use Weierstrass’ criterion of uniform convergence).  Since the limit function is

 $f(x)\;:=\;\lim_{n\to\infty}\left(x-\frac{x}{(1\!+\!x^{2})^{n}}\right)\;=\;x% \quad\forall x\in[-1,\,1],$

we have

 $\sup_{[-1,\,1]}|f_{n}(x)-f(x)|\;=\;\sup_{[-1,\,1]}\frac{|x|}{(1\!+\!x^{2})^{n}% }\,\longrightarrow 0\quad\mbox{as}\;\;n\to\infty,$

which means by Theorem 1 of the parent entry (http://planetmath.org/LimitFunction) that the sequence (1) converges uniformly on the interval to the identity function.  Further, the members of the sequence are continuous and differentiable.  Furthermore,

 $f_{n}^{\prime}(x)\;=\;1-\frac{1\!+(1\!-\!2n)x^{2}}{(1\!+\!x^{2})^{n+1}},$

whence

 $\lim_{n\to\infty}f_{n}^{\prime}(x)\;=\;1\quad(x\;\neq\;0).$

But in the point  $x=0$  we have

 $\lim_{n\to\infty}f_{n}^{\prime}(0)\;=\;\lim_{n\to\infty}0\;=\;0,$

which says that the limit of derivative sequence of (1) is discontinuous in the origin.  Because

 $f^{\prime}(x)\;\equiv\;1,$

we may write

 $\lim_{n\to\infty}\frac{d}{dx}f_{n}\;\neq\;\frac{d}{dx}\lim_{n\to\infty}f_{n}.$
Title derivative of limit function diverges from limit of derivatives DerivativeOfLimitFunctionDivergesFromLimitOfDerivatives 2013-03-22 19:00:29 2013-03-22 19:00:29 pahio (2872) pahio (2872) 7 pahio (2872) Example msc 40A30 msc 26A15 limit of derivatives diverges from derivative of limit function GrowthOfExponentialFunction