MCQ
$\mathop {lim}\limits_{n \to \infty } \cos \left( {\pi \sqrt {{n^2} + n} } \right)$ , where $n \in I$ , is
  • A
    $1$
  • B
    $-1$
  • $0$
  • D
    Does not exist

Answer

Correct option: C.
$0$
c
$\mathop {\lim }\limits_{n \to \infty }  \pm \cos \left( {n\pi  - \pi \sqrt {{n^2} + n} } \right)$

$\mathop {\lim }\limits_{n \to \infty }  \pm \cos \pi \left[ {\frac{{{n^2} - \left( {{n^2} + n} \right)}}{{n + \sqrt {{n^2} + n} }}} \right]$

$\mathop {\lim }\limits_{n \to \infty }  \pm \cos \left( {\pi  \times \frac{{ - n}}{{n + \sqrt {{n^2} + n} }}} \right)$

$ \pm \cos \left( {\pi  \times \frac{{ - 1}}{2}} \right) = 0$

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