Poly fun

$f(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+...+a_{0}$

[x]=m

$f(m)=a_{n}m^{n}+a_{n-1}m^{n-1}+...+a_{0}$

$f(f(m)+1)=a_{n}(f(m)+1)^{n}+a_{n-1}(f(m)+1){}^{n-1}+...+a_{0}=Cf(m)+f(1)$

$\frac{f(f(m)+1)}{f(m)}=C+\frac{f(1)}{f(m)}=integer$

m=1

$(x-a)(x-b)(x-c)-1=p(x)q(x)$ ......(1)

let p(x) is linear and q(x) is quadratic with integral coefficients

$p(a)q(a)=p(b)q(b)=p(c)q(c)=-1$

as p(a) is has integer coefficients and a is an integer

similarly with b and c

$p(a)=1or-1$

similar with b and c

a linear polynomial being one one cannot attain one- value for two numbers

therefore (1) as irrational roots

therefore the given limit =0 as $x_{i}\rightarrow irrational$

a=0 b=2