lim x → π / 4 ln ⁡ ( tan ⁡ ( x ) ) x − π / 4 Could you help me finding the limit?

Question

lim          x      →      π              /            4                  ln      ⁡      (      tan      ⁡      (      x      )      )              x      −      π              /            4      Could you help me finding the limit?

Lana Schwartz · Accepted Answer

Set             π      4        −  x  =  h to get  −      lim          h      →      0                          ln        ⁡        tan        ⁡                  (                                    π              4                                −          h          )                    h        =      lim          h      →      0                          ln        ⁡        (        1        +        tan        ⁡        h        )            h        −      lim          h      →      0                          ln        ⁡        (        1        −        tanh        )            h      Now any constant a,      lim          h      →      0                          ln        ⁡        (        1        +        a        tanh        )            h        =  a      lim          h      →      0                          ln        ⁡        (        1        +        a        tan        ⁡        h        )                    a        tan        ⁡        h              ⋅                    tan        ⁡        h            h        =  ⋯  =  a

myntfalskj4 · Accepted Answer

Let   f  (  x  )  =  ln  ⁡  (  tan  ⁡  x  )  .. We are looking at            f      (      x      )              x      −      π              /            4        =            f      (      x      )      −      f      (      π              /            4      )              x      −      π              /            4        .By definition the limit of this as   x  →  π      /    4 is just       f    ′    (  π      /    4  )  ,, an easy computation

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