Is it possible to find the following limit without using l'Hôpital's rule? lim x → π 4 cos x - 1 2 x - π 4

Question

Is it possible to find the following limit without using l&#039;Hôpital&#039;s rule?             lim              x        →                  π          4                                              cos          x                -                  1                      2                                      x        -                  π          4

Irene Tate · Accepted Answer

One way is to let       u    =    x    -          π      4      Step 1Thus we want             lim              u        →        0                                      cos                      (            u            +                          π              4                        )                          -                  1                      2                              u      Step 2Expand             cos              (        u        +                  π          4                )            , regroup and use the fundamental trigonometric limits.            lim              u        →        0                                      cos                      (            u            +                          π              4                        )                          -                  1                      2                              u        =          lim              u        →        0                                      1                      2                                    cos          u                -                  1                      2                                    sin          u                -                  1                      2                              u            =          lim              u        →        0                    (              1                  2                            (                                            cos              u                        -            1                    u                )            -              1                  2                            (                              sin            u                    u                )            )            =          1              2                    (      0      )        -          1              2                    (      1      )        =    -          1              2

Ciara Perez · Accepted Answer

Solution:Making       h    =    x    -          π      4       and using the definition of differential, we have            lim              x        →                  π          4                                              cos          x                -                  1                      2                                      x        -                  π          4                      ≡          lim              h        →        0                                      cos                      (                          π              4                        +            h            )                          -                  cos                      (                          π              4                        )                              h        =    -          sin              (                  π          4                )              =    -          1              2            Answer:       -          1              2

Is it possible to find the following limit without using l'Hôpital's rule? lim_(x rightarrow pi/4)(cos x -1/(sqrt2))/(x-pi/4)

Answered question

Answer & Explanation

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