` Lim_( x -> 0) ((1+x)^(1/k) - 1) / x` ,( k is positive integer)

Evaluate int_(0)^(1)(tx+1-x)^(n)dx , where n is a positive integer and t is a parameter independent of x . Hence , show that int_0^1 ​ x^k (1−x)^(n−k) dx= P/([.^nC_k(n+1)]) ​ for k=0,1,......n , then P=

Evaluate int_(0)^(1)(tx+1-x)^(n)dx , where n is a positive integer and t is a parameter independent of x . Hence , show that ∫ 0 1 ​ x k (1−x) n−k dx= [ n C k ​ (n+1)] P ​ fork=0,1,......n, then P=

lim_(x to 0)[(1+3x)^(1//x)]=k , then k is

Let f(x)=(a_(2k)x^(2k)+a_(2k-1)x^(2k-1)+...+a_(1)x+a_(0))/(b_(2k)x^(2k)+b_(2k-1)x^(2k-1)+...+b_(1)x+b_(0)) , where k is a positive integer, a_(i), b_(i) in R " and " a_(2k) ne 0, b_(2k) ne 0 such that b_(2k)x^(2k)+b_(2k-1)x^(2k-1)+...+b_(1)x+b_(0)=0 has no real roots, then

Let f(x)=(a_(2k)x^(2k)+a_(2k-1)x^(2k-1)+...+a_(1)x+a_(0))/(b_(2k)x^(2k)+b_(2k-1)x^(2k-1)+...+b_(1)x+b_(0)) , where k is a positive integer, a_(i), b_(i) in R " and " a_(2k) ne 0, b_(2k) ne 0 such that b_(2k)x^(2k)+b_(2k-1)x^(2k-1)+...+b_(1)x+b_(0)=0 has no real roots, then

f(x)=x^(2)+16 . For what value of k is f(2k+1)=2f(k)+1 if k is a positive integer?

lim_(xtooo)(1+k/x)^(m/x)

The least integral value of 'k' If (k-1) x ^(2) -(k+1) x+ (k+1) is positive for all real value of x is: