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Limit : Trigonometric Function

Evaluate limxπ4sinxcosxxπ4.

Solution (1)

           limxπ4sinxcosxxπ4

        = limxπ4(sinxcosxxπ4×2222)

        = limxπ422sinx22cosx22(xπ4)

        = limxπ422sinx22cosx22(xπ4)

        = limxπ4sinxcosπ4cosxsinπ422(xπ4)

        = 2lim(xπ4)0sin(xπ4)(xπ4)

        =2×1

        =2

Solution (2)

         Let t=xπ4 and hence x=t+π4

         When xπ4, t0.

              limxπ4sinxcosxxπ4 

          =limt0sin(t+π4)cos(t+π4)t 

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         =limt0sintcosπ4+costsinπ4costcosπ4+sintcosπ4t 

         =limt02sintcosπ4t 

         =limt02sint(22)t 

         =2limt0sintt 

         =2×1 

         =2   

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