lim x tends to 0 (5 tanx sinx upon x square) here ( ) this bracket indicates greatest integer function

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You can calculate the limit of this function using basic concept of LHL=RHL. First try for x=0.01 and -0.01. The function reduces to same in both cases. So it is symmetric around x=0. Now the limit of inner function is 5 in both cases. So the limit exists and it is equal to 5.