(1+(1)/(x))^(x)+x^(1+(1/x))

If y=(1+(1)/(x))^(x), then (dy)/(dx)=(1+(1)/(x))^(x){log(1+(1)/(x))-(1)/(x+1)}(b)(1+(1)/(x))^(x)log(1+(1)/(x))(c)(x+(1)/(x))^(x){log(x+1)-(x)/(x+1)}(d)(x+(1)/(x))^(x){log(1+(1)/(x))+(1)/(x+1)}

For x>0 the sum of the series (1)/(1+x)-(1-x)/((1+x)^(2))+((1-x)^(2))/((1+x)^(3))-...oo is equal to

The expression (1)/(1+(x)/(1-(x)/(x+2)))-:((1)/(1-x)+(1)/(1+x))/((1)/(1-x)-(1)/(1+x))

The expressin (1)/(1+(x)/(1-(x)/(x+2)))div ((1)/(1-x)+(1)/(1+x))/((1)/(1-x)-(1)/(1+x))

lim_(x rarr1)((1+x)/(2+x))^((1-sqrt(x))/(1-x))

f(x)={(e^((1)/(x-1)))/((1+e^(((1)/(x-1))),x!=1}x-1;x=1)

For x gt 0 the sum of the series (1)/(1+x) - ((1-x))/((1+x)^(2))+ ((1-x)^(2))/((1+x)^(3))- cdots oo is equal to

If x is positive,the sum to infinity of the series (1)/(1+x)-(1-x)/((1+x)^(2))+((1-x)^(2))/((1+x)^(3))-((1-x)^(3))/((1+x)^(4))+is1/2 (b) 3/4 (c) 1 (d) none of these

If x is positive, the sum to infinity of the series 1/(1+x)-(1-x)/((1+x)^2)+(1-x) ^2/((1+x)^3)-(1-x)^3/((1+x)^4)+....... i s (a)1/2 (b) 3/4 (c) 1 (d) none of these