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

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=λx-1x-3x-5+2x+1⇒  limx→∞  x2λx-1x-3x-5+2x+12x5+3=6⇒    λ2=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=λx-1x-3x-5+2x+1⇒  limx→∞  x2λx-1x-3x-5+2x+12x5+3=6⇒    λ2=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