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Math Help - Functions and limits?

  1. #1
    Super Member fardeen_gen's Avatar
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    Functions and limits?

    If 2f(\sin x) + \sqrt{2}f(\cos x) = \tan x then evaluate :

    \lim_{x\rightarrow 1} \sqrt{(1 - x)}f(x)
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  2. #2
    Senior Member
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    Find f(x) explicitly

    Rewrite the condition as: 2f(a)+\sqrt2f(b)=\frac ab and 2f(b)+\sqrt2f(a)=\frac ba for all a^2+b^2=1. Using a 2x2 matrix to solve, we get f(a)=\frac ab-\frac{\sqrt2}2\frac ba and f(b)=\frac ba-\frac{\sqrt2}2\frac ab, so f(\sin x)=\tan x-\frac{\sqrt2}2\cot x and f(\cos x)=\cot x-\frac{\sqrt2}2\tan x. Rewriting another way, f(x)=\frac x{\sqrt{1-x^2}}-\frac{\sqrt2}2\frac{\sqrt{1-x^2}}x

    So \lim_{x\to 1}\sqrt{1-x}f(x)=\lim_{x\to1}\frac x{\sqrt{1+x}}-\frac{\sqrt2}2\frac{\sqrt{1+x}(1-x)}x=\frac{\sqrt2}2
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