- #1

funcalys

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Is there a function [itex]f(x): \mathbb{R} \to \mathbb{R}[/itex] such that [itex]\lim_{x \to 0} x f(x) = a \neq 0[/itex].

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- Thread starter funcalys
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- #1

funcalys

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- #2

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What can you tell about ##\lim_{x\rightarrow 0} f(x)##. Is it possible that this limit is finite?

- #3

CompuChip

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If f(x) = 1/x then x f(x) = 1, except that doesn't work in x = 0. But maybe that will get you started.

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- #4

CompuChip

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In the end the answer is: yes, there is a function [itex]f(x): \mathbb{R} \to \mathbb{R}[/itex] such that [itex]\lim_{x \to 0} x f(x) = a \neq 0[/itex].

- #5

Tosh5457

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Write xf(x) + ε(x) = g(x)

with ε(x) and g(x) such as

lim (x->0) ε(x) = 0 and lim (x->0) g(x) = a

so that lim (x->0) ( xf(x) + ε(x) = g(x) ) <=> lim (x->0) xf(x) = a

Now define a ε(x) and g(x) that satisfy those conditions and you can find a function f(x) that satisfies that limit.

If you want x = 0 to be in the domain of f(x), you should also set the conditions ε(0) = 0 and g(0) = 0.

- #6

HallsofIvy

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