Calculate lim x → π / 2 cos ⁡ x x − π 2 by relating it to a value of ( cos ⁡ x ) ′

Question

Calculate      lim          x      →      π              /            2                          cos        ⁡        x                    x        −                  π          2                    by relating it to a value of   (  cos  ⁡  x      )    ′

humusen6p · Accepted Answer

First, we note that   cos  ⁡  π      /    2  =  0. The derivative       f    ′    (      x    0    ) of a function f at the point       x    0   is given by                    (1)                              f          ′                (                  x          0                )        =                  lim                      x            →                          x              0                                                            f            (            x            )            −            f            (                          x              0                        )                                x            −                          x              0                                          Now, we have                    (2)                              lim                      x            →            π                          /                        2                                                cos            ⁡            (            x            )                                x            −            π                          /                        2                          =                  lim                      x            →            π                          /                        2                                                cos            ⁡            (            x            )            −            cos            ⁡            (            π                          /                        2            )                                x            −            π                          /                        2                              Comparing (1) and (2), we see that if   f  (  x  )  =  cos  ⁡  x and       x    0    =  π      /    2, then                              d                    cos          ⁡          x                          d          x                    =              lim                  x          →          π                      /                    2                                      cos          ⁡          (          x          )                          x          −          π                      /                    2                    =      −      sin      ⁡      (      π              /            2      )      =      −      1

gvaldytist · Accepted Answer

L&#039;Hôpital&#039;s Rule relates the limit to the derivative of cosx, since substitution of   π      /    2 in the original limit yields   0      /    0Taking the derivative of each of the numerator and the denominator gives   −  sin  ⁡  x. We can simply plug in   π      /    2 and obtain −1.      lim          x      →      π              /            2                          cos        ⁡        x                    x        −                  π          2                      =      lim          x      →      π              /            2                          d                  /                d        x        (        cos        ⁡        x        )                    d                  /                d        x        (        x        −                  π          2                )              =      lim          x      →      π              /            2        (  −  sin  ⁡  x  )  =      lim          x      →      π              /            2        (  −  sin  ⁡  (      π    2    )  )  =  −  1

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