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Math Help - Limit?!

  1. #1
    Super Member dhiab's Avatar
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    Limit?!

    Calculate this limit :

    \mathop {\lim }\limits_{x \to + \infty } \left( {\sin \left( {\pi \sqrt {x^2 + x} } \right)} \right)
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  2. #2
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    Quote Originally Posted by dhiab View Post
    Calculate this limit :

    \mathop {\lim }\limits_{x \to + \infty } \left( {\sin \left( {\pi \sqrt {x^2 + x} } \right)} \right)

    The limit doesn't exist: take the sequence \{x_n\}=\left\{\,\frac{-1+\sqrt{4n^2+1}}{2}\,\right\} \Longrightarrow \sqrt{x_n^2+x_n}=n\Longrightarrow \sin(\pi\sqrt{x_n^2+x_n})=\sin n\pi=0 , and now

    take the sequence \{y_n\}=\left\{\frac{-1+\sqrt{n^2+1}}{2}\right\}\Longrightarrow \sqrt{y_n^2+y_n}=\frac{n}{2} \Longrightarrow \sin(\pi\sqrt{y_n^2+y_n})=\sin\frac{n}{2}\pi=\pm 1

    Tonio

    Ps. In fact it is enough to take the sequence \{y_n\} ...
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