Limit as x → 2 of cos ⁡ ( π x ) x − 2

Question

Limit as   x  →  2 of             cos      ⁡      (              π        x            )              x      −      2

Jayvion Mclaughlin · Accepted Answer

HINT:  cos  ⁡            π      x        =  sin  ⁡      (                  π        2              −                  π        x              )    =  sin  ⁡                    π        (        x        −        2        )                    2        x            Now set                     π        (        x        −        2        )                    2        x              =  y and use       lim          h      →      a                          sin        ⁡        (        h        −        a        )                    h        −        a              =  1

Jameson Lucero · Accepted Answer

Let             1      x        =  y then                              lim                      x            →            2                                                cos            ⁡                                          (                                  π                  x                                )                                                          x            −            2                                              =                  lim                      y            →                          1              2                                                            sin            ⁡                                          (                                  π                  2                                −                π                y                )                                                                                        1                y                                      −            2                                                            =                  lim                      y            →                          1              2                                                            sin            ⁡                          π                              (                                  1                  2                                −                y                )                                                          π                          (                              1                2                            −              y              )                                                                          (                              1                2                            −              y              )                                      1              −              2              y                                      π        y                                          =                                            π              4

Limit as x → 2 of cos ⁡

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