$$Letf(x)$$ be a polynomial satisfying limx→∞$$x2fx2x5+3=6$$ and $$f1=3$$, $$f3=7$$ and $$f5=11$$. thenThe value of $$f(0)$$ islimx→$$1x-1sinfx-2x-1$$ is equal to

Question

$$Letf(x)$$ be a polynomial satisfying limx→∞$$x2fx2x5+3=6$$ and $$f1=3$$, $$f3=7$$  and  $$f5=11$$. thenThe value of  $$f(0)$$ islimx→$$1x-1sinfx-2x-1$$  is equal to

Infinity Learn · Accepted Answer

Since limx→∞ $$x2fx2x5+3=6$$,  which  is  finite  $$non-zero$$,  fx  must  be  polynomial  of  degree  '$$3$$'.Also  $$f1=3$$,  $$f3=7$$  and  $$f5=11$$∴     $$fx=$$λ$$x-1x-3x-5+2x+1$$⇒  limx→∞  $$x2$$λ$$x-1x-3x-5+2x+12x5+3=6$$⇒    λ$$2=6$$⇒  λ$$=12$$∴   $$fx=12x-1x-3x-5+2x+1So$$,   $$f0=12-1-3-5+1=-179$$     limx→$$1$$   $$x-1sinfx-2x-1$$       $$=$$  limx→$$1$$  $$x-1sin12x-1x-3x-5$$       $$=$$  limx→$$1$$  $$12x-1x-3x-5sin12x-1x-3x-5.112x-3x-5$$         $$=$$  $$196$$ Since limx→∞ $$x2fx2x5+3=6$$,  which  is  finite  $$non-zero$$,  fx  must  be  polynomial  of  degree  '$$3$$'.Also  $$f1=3$$,  $$f3=7$$  and  $$f5=11$$∴     $$fx=$$λ$$x-1x-3x-5+2x+1$$⇒  limx→∞  $$x2$$λ$$x-1x-3x-5+2x+12x5+3=6$$⇒    λ$$2=6$$⇒  λ$$=12$$∴   $$fx=12x-1x-3x-5+2x+1So$$,   $$f0=12-1-3-5+1=-179$$     limx→$$1$$   $$x-1sinfx-2x-1$$       $$=$$  limx→$$1$$  $$x-1sin12x-1x-3x-5$$       $$=$$  limx→$$1$$  $$12x-1x-3x-5sin12x-1x-3x-5.112x-3x-5$$         $$=$$  $$196$$

Letf(x) be a polynomial satisfying limx→∞x2fx2x5+3=6 and f1=3, f3=7 and f5=11. thenThe value of f(0) islimx→1x-1sinfx-2x-1 is equal to

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