We must to calculate lim x → π / 2 sin ⁡ x − 1 2 x − π .

Question

We must to calculate      lim          x      →      π              /            2                  sin      ⁡      x      −      1              2      x      −      π        .

Zayden Wiley · Accepted Answer

We must to calculate      lim          x      →      π              /            2                  sin      ⁡      x      −      1              2      x      −      π      If we use a change of variables      h  =  x  −      π    2    ,    then this becomes:      lim          h      →             0                  sin      ⁡              (                  π          2                +        h        )            −      sin      ⁡              (                  π          2                )                    2      h            =          (      ∗      )            1    2           f    ′        (          π      2        )    ,where      f  (  x  )  =  sin  ⁡  (  x  )  .    All we have done at (*) is use the definition of          f    ′    (  x  )  .But          f    ′        (          π      2        )    =  0  ,    and therefore the answer is          1    2    ×  0  =  0.Also, from the comments, you went wrong here:      lim          t      →      0                  sin      ⁡              (        t        +                  π          2                )            −      sin      ⁡              π        2                    2      t        =      lim          t      →      0                  2      sin      ⁡              (        t        +        π        )            cos      ⁡      t              2      t      This is incorrect. The correct formula is:     sin  ⁡  α  −  sin  ⁡  β  =  2  cos  ⁡            α      +      β        2    sin  ⁡            α      −      β        2    ,    which would get you:     sin  ⁡      (    t    +          π      2        )    −  sin  ⁡      (          π      2        )    =  2  cos  ⁡      (          1      2              (      t      +      π      )        )    sin  ⁡      (          1      2        t    )

Yesenia Sherman · Accepted Answer

Another way to rewrite it is            sin      ⁡      (      t      +      π              /            2      )      −      1              2      t        =            sin      ⁡      t      cos      ⁡      (      π              /            2      )      +      sin      ⁡      (      π              /            2      )      cos      ⁡      t      −      1              2      t        =            cos      ⁡      t      −      1              2      t        .

We must to calculate <munder> <mo movablelimits="true" form="prefix">lim <mrow class="MJ

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New Questions in Calculus 1