Let $$fx=$$$$sin$$$$-11-x$$×$$cos$$$$-11-x2x$$×$$1-x$$, where $${x}$$ denotes the fractional part of x. $$R=limx$$→$$0+$$ fx is equal to $$L=limx$$→$$0-$$ fx is equal toWhich of the following is true?

Question

Let $$fx=$$$$sin$$$$-11-x$$×$$cos$$$$-11-x2x$$×$$1-x$$, where $${x}$$ denotes the fractional part of x. $$R=limx$$→$$0+$$  fx  is equal to $$L=limx$$→$$0-$$  fx is equal toWhich of the following is true?

Infinity Learn · Accepted Answer

Let  $$fx=$$$$sin$$$$-11-x$$×$$cos$$$$-11-x2x$$×$$1-x$$∴   limx→$$0+$$  fx  $$=$$  limh→$$0$$  $$f0+h$$         $$=limh$$→$$0$$  $$sin$$$$-11-0+hcos-11-0+h20+h1-0+h$$         $$=$$ limh→$$0$$  $$sin$$$$-11-hcos-11-h2h1-h$$         $$=limh$$→$$0$$  $$sin$$$$-11-h1-h$$   limh→$$0$$  $$cos$$$$-11-h2hIn$$   second   limit,  put   $$1-h$$  $$=$$  $$cos$$θ.  Thenlimx→$$0+$$  $$fx=limh$$→$$0$$  $$sin$$$$-11-h1-h$$  limθ→$$0$$   $$cos$$$$-1cos$$θ$$21-$$$$cos$$θ                   $$=limh$$→$$0$$  $$sin$$$$-11-h1-h$$    limθ→$$0$$  θ$$2sin$$θ$$2$$ $$=$$π$$2$$    θ >$$0and$$    limx→$$0-$$  fx  $$=$$  limh→$$0$$  $$f0-h$$                         $$=$$  limh→$$0$$  $$sin$$$$-11-0-hcos-11-0-h20-h1-0-h$$                          $$=$$  limh→$$0$$  $$sin$$$$-11+h-1cos-11+h-12-h+11+h-1$$                             $$=limh$$→$$0$$  $$sin$$$$-1hh$$  ..  $$cos$$$$-1h21--h$$                            $$=$$  $$1$$$$π$$$$22$$  $$=$$π$$22Let$$ $$fx=$$$$sin$$$$-1x-x$$×$$cos$$$$-11-x2x$$×$$1-x$$∴   limx→$$0+$$  fx  $$=$$  limh→$$0$$  $$f0+h$$         $$=limh$$→$$0$$  $$sin$$$$-110+hcos-11-0+h20+h1-0+h$$         $$=$$ limh→$$0$$  $$sin$$$$-11-hcos-11-h2h1-h$$         $$=limh$$→$$0$$  $$sin$$$$-11-h1-h$$   limh→$$0$$  $$cos$$$$-11-h2hIn$$   second   limit,  put   $$1-h$$  $$=$$  $$cos$$θ.  Thenlimx→$$0+$$  $$fx=limh$$→$$0$$  $$sin$$$$-11-h1-h$$  limθ→$$0$$   $$cos$$$$-1cos$$θ$$21-$$$$cos$$θ                   $$=limh$$→$$0$$  $$sin$$$$-11-h1-h$$    limθ→$$0$$  θ$$2sin$$θ$$2$$     θ >$$0and$$    limx→$$0+$$  fx  $$=$$  limh→$$0$$  $$f0-h$$                         $$=$$  limh→$$0$$  $$sin$$$$-11-0-hcos-11-0-h20-h1-0-h$$                          $$=$$  limh→$$0$$  $$sin$$$$-11+h-1cos-11+h-12-h+11+h-1$$                             $$=limh$$→$$0$$  $$sin$$$$-1hh$$  ..  $$cos$$$$-1h21--h$$                            $$=$$  $$1$$$$π$$$$22$$  $$=$$π$$22Let$$ $$fx=$$$$sin$$$$-1x-x$$×$$cos$$$$-11-x2x$$×$$1-x$$∴   limx→$$0+$$  fx  $$=$$  limh→$$0$$  $$f0+h$$         $$=limh$$→$$0$$  $$sin$$$$-110+hcos-11-0+h20+h1-0+h$$         $$=$$ limh→$$0$$  $$sin$$$$-11-hcos-11-h2h1-h$$         $$=limh$$→$$0$$  $$sin$$$$-11-h1-h$$   limh→$$0$$  $$cos$$$$-11-h2hIn$$   second   limit,  put   $$1-h$$  $$=$$  $$cos$$θ.  Thenlimx→$$0+$$  $$fx=limh$$→$$0$$  $$sin$$$$-11-h1-h$$  limθ→$$0$$   $$cos$$$$-1cos$$θ$$21-$$$$cos$$θ                   $$=limh$$→$$0$$  $$sin$$$$-11-h1-h$$    limθ→$$0$$  θ$$2sin$$θ$$2$$     θ >$$0and$$    limx→$$0+$$  fx  $$=$$  limh→$$0$$  $$f0-h$$                         $$=$$  limh→$$0$$  $$sin$$$$-11-0-hcos-11-0-h20-h1-0-h$$                          $$=$$  limh→$$0$$  $$sin$$$$-11+h-1cos-11+h-12-h+11+h-1$$                             $$=limh$$→$$0$$  $$sin$$$$-1hh$$  ..  $$cos$$$$-1h21--h$$                            $$=$$  $$1$$$$π$$$$22$$  $$=$$π$$22$$

Let fx=sin-11-x×cos-11-x2x×1-x, where {x} denotes the fractional part of x. R=limx→0+ fx is equal to L=limx→0- fx is equal toWhich of the following is true?

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