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Math Help - continuous at infinty?

  1. #1
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    Question continuous at infinty?

    show that f is continuous at (-infinity, infinity)

    f(x)= { sin(x) if x < (pi/4)
    cos(x) if x >or equal to (pi/4)



    confused???
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  2. #2
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    Quote Originally Posted by thecount View Post
    show that f is continuous at (-infinity, infinity)

    f(x)= { sin(x) if x < (pi/4)
    cos(x) if x >or equal to (pi/4)



    confused???
     f\left( x \right) = \left\{ \begin{gathered}<br />
  \sin x,{\text{    if  }}x < \frac{\pi }<br />
{4} \hfill \\<br />
 \cos x,{\text{   if  }}x \geqslant \frac{\pi }<br />
{4} \hfill \\ <br />
\end{gathered}  \right. \hfill \\

     f\left( x \right){\text{  is continuous in }}\left( { - \infty ,{\text{ }}\infty } \right){\text{ if it is continuous at }}x = \frac{\pi }<br />
{4}. \hfill \\

      {\text{For  }}f\left( x \right){\text{ to be continuous at }}x = \frac{\pi }<br />
{4}, \hfill \\

     \mathop {\lim }\limits_{x \to \frac{\pi }<br />
{4} - } f\left( x \right) = \mathop {\lim }\limits_{x \to \frac{\pi }<br />
{4} + } f\left( x \right) = f\left( {\frac{\pi }<br />
{4}} \right) \hfill \\
    <br />
  {\text{Now, Left hand limit, }} = \mathop {\lim }\limits_{x \to \frac{\pi }<br />
{4} - } f\left( x \right) = \mathop {\lim }\limits_{x \to \frac{\pi }<br />
{4} - } \sin x = \sin \frac{\pi }<br />
{4} = \frac{1}<br />
{{\sqrt 2 }} \hfill \\

     {\text{Right hand limit, }} = \mathop {\lim }\limits_{x \to \frac{\pi }<br />
{4} + } f\left( x \right) = \mathop {\lim }\limits_{x \to \frac{\pi }<br />
{4} + } \cos x = \cos \frac{\pi }<br />
{4} = \frac{1}<br />
{{\sqrt 2 }} \hfill \\

      f\left( {\frac{\pi }<br />
{4}} \right) = \cos \frac{\pi }<br />
{4} = \frac{1}<br />
{{\sqrt 2 }} \hfill \\

     {\text{Since, }}\mathop {\lim }\limits_{x \to \frac{\pi }<br />
{4} - } f\left( x \right) = \mathop {\lim }\limits_{x \to \frac{\pi }<br />
{4} + } f\left( x \right) = f\left( {\frac{\pi }<br />
{4}} \right) = \frac{1}<br />
{{\sqrt 2 }}\hfill \\

     {\text{So, the function is continuous at }}x = \frac{\pi }<br />
{4} \hfill \\

      {\text{So, the function  }}f\left( x \right){\text{  is continuous in }}\left( { - \infty ,{\text{ }}\infty } \right) \hfill \\ <br />
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