If `sum_(r=1)^kcos^(- 1)beta_r=(kpi)/2` for any `k >= 1 and A = sum_(r=1)^k(beta_r)^r` then `lim_(x->A)((1+x)^(1/3)-(1-2x)^(1/4))/(x+x^2)` is equal to

If sum_(r = 1)^(k) cos^(-1) beta r = (kpi)/(2), "for any k" ge 1 and A = sum_(r = 1)^(k)(beta r)^(r),"then" lim_(x rarr A) ((1 + x^(2))^(1//3)-(1-2x)^(1//4))/(x+x^(2)) is equal to :

Let 0lt=beta_(r)lt=1 and sum_(r=1)^(k)cos^(-1)beta_(r)=(k pi)/(2) for any kgt=1 and A=sum_(r=1)^(k)(beta_(r))^(r) ,then the number of solutions of the equation tan^(2)x-sec^(10)x+1=A in (0,A+10) is/are (1) 3 (2)2 (3) 1 (4)zero

lim_(xrarr2)(sum_(r=1)^(n)x^r-sum_(r=1)^(n)2r)/(x-2) is equal to

lim_(xrarr2)(sum_(r=1)^(n)x^r-sum_(r=1)^(n)2^r)/(x-2) is equal to

lim_(xrarr1) (sum_(x=1)^(n)x^(r)-n)/(x-1) is equal to

lim_(xto1) (sum_(r=1)^(n)x^(r)-n)/(x-1) is equal to