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单词 DerivativeOfLimitFunctionDivergesFromLimitOfDerivatives
释义

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

limnddxfn(x)ddxlimnfn(x),

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

Example.  The function sequence

fn(x):=j=1nx3(1+x2)j=x-x(1+x2)n  (n= 1, 2, 3,)(1)

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

x31+x2+x3(1+x2)2+x3(1+x2)2+

(although one cannot use Weierstrass’ criterion of uniform convergenceMathworldPlanetmath).  Sincethe limit function is

f(x):=limn(x-x(1+x2)n)=xx[-1, 1],

we have

sup[-1, 1]|fn(x)-f(x)|=sup[-1, 1]|x|(1+x2)n0asn,

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,

fn(x)= 1-1+(1-2n)x2(1+x2)n+1,

whence

limnfn(x)= 1(x 0).

But in the point  x=0  we have

limnfn(0)=limn0= 0,

which says that the limit of derivativePlanetmathPlanetmath sequence of (1) is discontinuousMathworldPlanetmath in the origin.  Because

f(x) 1,

we may write

limnddxfnddxlimnfn.
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更新时间:2025/5/4 23:29:17