If `alpha in(0,1)` and `f:R->R` and `lim_(x->oo)f(x)=0,lim_(x->oo)(f(x)-f(alphax))/x=0,` then `lim_(x->oo)f(x)/x=lambda` where `2lambda+7` is

lim_(x rarr oo) (1+f(x))^(1/f(x))

If f(x) = sqrt((x-sinx)/(x+cos^(2)x)) , then lim_(x rarr oo) f(x) =

Let f(x)=lim_(n rarr oo)(cos x)/((1+tan^(-1)x)^(n)), then int_(0)^(oo)f(x)dx=

Given f(x)=(ax+b)/(x+1), lim_(x to oo)f(x)=1 and lim_(x to 0)f(x)=2 , then f(-2) is

If f(x)=(x^(2))/(1+x^(2)), prove that lim_(x rarr oo)f(x)=1

If lim_(x->oo) f(x) exists and is finite and nonzero and if lim_(x->oo) {f(x)+(3f(x)−1)/(f^2(x))}=3 ,then the value of lim_(x->oo) f(x) is

Sketch the graph of an example of a rational function f that satisfies all the given conditions. (i) f(0)=0,f(1), lim_(xrarroo) f(x)=0,f " is odd" (ii) lim_(xrarr0^(+)) f(x)=oo,lim_(xrarr0^(-)) f(x)=-oo,lim_(xrarroo) f(x)=1,lim_(xrarr-oo) f(x)=1 (iii) lim_(xrarr-2) f(x)=oo, lim_(xrarr-oo) f(x)=3,lim_(xrarroo) f(x) = 3 , f(0)=0

Let f :R to [0,oo) be such that lim_(xto3) f(x) exists and lim_(x to 3) ((f(x))^2-4)/sqrt(|x-3|)=0 . Then lim_(x to 3) f(x) equals

If lm_(x->oo) f(x) exists and is finite and nonzero and if lim_(x->oo) {{f(x)+(3f(x)−1)/(f_2(x))}=3 ,then the value of lim_(x->oo) f(x) is

If f(x)=lim_(m->oo) lim_(n->oo)cos^(2m) n!pix then the range of f(x) is