Evaluate: lim θ → π 4 cos ⁡ θ − sin ⁡ θ θ − π 4

Question

Evaluate:       lim          θ      →              π        4                                cos        ⁡        θ        −        sin        ⁡        θ                    θ        −                  π          4

Kroatujon3 · Accepted Answer

Add and subtract   cos  ⁡      (          π      4        )    =  sin  ⁡      (          π      4        )   in the numerator to get:      lim          θ      →                        π          4                                        cos        ⁡        θ        −        sin        ⁡        θ                    θ        −                              π            4                                =      lim          θ      →                        π          4                                        cos        ⁡        θ        −        cos        ⁡                  (                      π            4                    )                −                  (          sin          ⁡          θ          −          sin          ⁡                      (                          π              4                        )                    )                            θ        −                              π            4                                =      lim          θ      →                        π          4                                        cos        ⁡        θ        −        cos        ⁡                  (                      π            4                    )                            θ        −                              π            4                                −      lim          θ      →                        π          4                                        sin        ⁡        θ        −        sin        ⁡                  (                      π            4                    )                            θ        −                              π            4                                =  −  sin  ⁡      (                  π        4              )    −  cos  ⁡      (                  π        4              )    =  −      2  where we used the definition for derivatives of sine and cosine.

Gauge Terrell · Accepted Answer

Let   θ  −      π    4    =  x. Hence,   θ  =      π    4    +  x andHence,      lim          θ      →                        π          4                                        cos        ⁡        θ        −        sin        ⁡        θ                    θ        −                              π            4                                =      lim          x      →      0                  cos      ⁡              (                  π          4                +        x        )            −      sin      ⁡              (                  π          4                +        x        )              x    =  −      2        lim          x      →      0                  sin      ⁡              x              x    =  −      2    .

Evaluate: <munder> <mo movablelimits="true" form="prefix">lim <mrow class="MJX-TeXAtom

Answered question

Answer & Explanation

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