Friday, 7 December 2012

real analysis - Use epsilondelta definition to prove limlimitsxrightarrowx0sqrt[3]f(x)=sqrt[3]A

It's known that limxx0f(x)=A, how to prove limxx03f(x)=3A?



Here's what I've got now:



When A=0, to prove limxx03f(x)=0: Since we have limxx0f(x)=A=0, so |f(x)|<ϵ. => |3f(x)|<ϵ30<ϵ



When A0, |3f(x)3A|=|f(x)A||f(x)23+(f(x)A)13+A23|...




How can I deal with (f(x)A)13? Thanks.

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