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Thread: 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???
    $\displaystyle f\left( x \right) = \left\{ \begin{gathered}
    \sin x,{\text{ if }}x < \frac{\pi }
    {4} \hfill \\
    \cos x,{\text{ if }}x \geqslant \frac{\pi }
    {4} \hfill \\
    \end{gathered} \right. \hfill \\$

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

    $\displaystyle {\text{For }}f\left( x \right){\text{ to be continuous at }}x = \frac{\pi }
    {4}, \hfill \\$

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

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

    $\displaystyle f\left( {\frac{\pi }
    {4}} \right) = \cos \frac{\pi }
    {4} = \frac{1}
    {{\sqrt 2 }} \hfill \\$

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

    $\displaystyle {\text{So, the function is continuous at }}x = \frac{\pi }
    {4} \hfill \\$

    $\displaystyle {\text{So, the function }}f\left( x \right){\text{ is continuous in }}\left( { - \infty ,{\text{ }}\infty } \right) \hfill \\
    $
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