Results 1 to 3 of 3

Math Help - Composition of functions

  1. #1
    Newbie
    Joined
    Jan 2009
    Posts
    2

    Composition of functions

    If we assume:

    <br />
\begin{gathered}<br />
  f\left( x \right) \to y_0 {\text{ as }}x \to x_0  \hfill \\<br />
  g\left( y \right) \to l{\text{ as }}y \to y_0  \hfill \\<br />
  g\left( {y_0 } \right) = l \hfill \\ <br />
\end{gathered} <br />
,

    how can I show that <br />
\mathop {\lim }\limits_{x \to x_0 } g\left( {f\left( x \right)} \right) = l<br />
?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Newbie
    Joined
    Jan 2009
    Posts
    2
    Any ideas? I'm really stuck here.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,605
    Thanks
    1573
    Awards
    1
    If \varepsilon  > 0 from the given \left( {\exists \delta  > 0} \right)\left[ {0 < \left| {y - y_0 } \right| < \delta \, \Rightarrow \,\left| {g(y) - l} \right| < \varepsilon } \right].
    Again \,\delta  > 0 \Rightarrow \,\left( {\exists \delta ' > 0} \right)\left[ {0 < \left| {x - x_0 } \right| < \delta '\, \Rightarrow \,\left| {f(x) - y_0 } \right| < \delta } \right].
    We are given that g\left( {y_0 } \right) = l. Define \delta '' = \min \left\{ {\delta ,\delta '} \right\}.

    0 < \left| {y - x_0 } \right| < \delta ''\, \Rightarrow \,\left| {f(y) - y_0 } \right| < \delta \, \Rightarrow \,\left| {g\left( {f(y)} \right) - l} \right| <\varepsilon

    Therefore: g \circ f(y) \to l\;,\,y \to x_0
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Composition of Two Functions
    Posted in the Algebra Forum
    Replies: 3
    Last Post: May 14th 2011, 11:26 AM
  2. Composition of Functions
    Posted in the Algebra Forum
    Replies: 2
    Last Post: February 22nd 2010, 07:49 PM
  3. Functions / Composition
    Posted in the Pre-Calculus Forum
    Replies: 2
    Last Post: November 12th 2008, 03:26 PM
  4. Composition of Functions
    Posted in the Pre-Calculus Forum
    Replies: 2
    Last Post: September 14th 2008, 04:41 PM
  5. Composition of Two Functions
    Posted in the Pre-Calculus Forum
    Replies: 1
    Last Post: January 9th 2008, 03:53 PM

Search Tags


/mathhelpforum @mathhelpforum