Results 1 to 2 of 2

Thread: Functions and limits?

  1. #1
    Super Member fardeen_gen's Avatar
    Joined
    Jun 2008
    Posts
    539

    Functions and limits?

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

    $\displaystyle \lim_{x\rightarrow 1} \sqrt{(1 - x)}f(x)$
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member
    Joined
    Apr 2009
    From
    Atlanta, GA
    Posts
    409

    Find f(x) explicitly

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

    So $\displaystyle \lim_{x\to 1}\sqrt{1-x}f(x)=\lim_{x\to1}\frac x{\sqrt{1+x}}-\frac{\sqrt2}2\frac{\sqrt{1+x}(1-x)}x=\frac{\sqrt2}2$
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Limits and functions
    Posted in the Calculus Forum
    Replies: 1
    Last Post: Sep 26th 2010, 06:29 PM
  2. Limits of functions
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: Apr 26th 2010, 07:41 PM
  3. Limits of Functions?
    Posted in the Calculus Forum
    Replies: 8
    Last Post: Oct 2nd 2009, 09:37 PM
  4. Limits of functions
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: Apr 9th 2009, 05:08 AM
  5. Limits of e functions
    Posted in the Calculus Forum
    Replies: 9
    Last Post: Sep 29th 2008, 11:20 AM

Search Tags


/mathhelpforum @mathhelpforum