Find lim x → 0 cos ⁡ ( π x 2 csc ⁡ ( x 2 ) cot ⁡ ( 6 x ) )

Question

Find       lim          x      →      0        cos  ⁡            (        π      x    2    csc  ⁡  (      x    2    )  cot  ⁡  (  6  x  )            )

Kristen Garrison · Accepted Answer

Note that  π      x    2    csc  ⁡  (      x    2    )  cot  ⁡  (  6  x  )  =      π    3    ⋅            x      2              sin      ⁡      (              x        2            )        ⋅            6      x              sin      ⁡      (      6      x      )        cos  ⁡  (  6  x  )  .Since       lim          x      →      0                  sin      ⁡      x        x    =  1 and       lim          x      →      0        cos  ⁡  (  x  )  =  1, we have      lim          x      →      0            x    2    csc  ⁡  (      x    2    )  cot  ⁡  (  6  x  )  =      1    3    .Since cos is a continuous function, we have      lim          x      →      0        cos  ⁡            (        π      x    2    csc  ⁡  (      x    2    )  cot  ⁡  (  6  x  )            )        =  cos  ⁡            (            lim          x      →      0        π      x    2    csc  ⁡  (      x    2    )  cot  ⁡  (  6  x  )            )        =  cos  ⁡      π    3    =      1    2    .

Find lim_(x => 0) cos(pi x^2 csc(x/2) cot(6x))

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Answer & Explanation

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