`lim_(xto0^(-)) (sum_(r=1)^(2n+1)[x^(r)]+(n+1))/(1+[x]+|x|+2x),` where `ninN` and `[.]` denotes the greatest integer function, equals

Answer

C

Explanation

`underset(xto0^(-))lim(sum_(r=1)^(2n+1)[x^(r)]+(n+1))/(1+[x]+|x|+2x)=underset(xto0^(-))lim(sum_(r=1)^(2n+1)[x^(r)]+(n+1))/(x)`
`=underset(xto0^(-))lim([x]+[x^(2)]+[x^(3)]+...+[x^(2n+1)]+(n+1))/(x)`
`=underset(xto0^(-))lim(ubrace((-1)+(-1)+...+(-1))_((n+1)"times")+(n+1))/(x)=underset(xto0^(-))lim(0)/(x)=0`