If f(x) is continuous in `[0,1] and f(1/2)=1.` prove that `lim_(n->oo)f((sqrt(n))/(2sqrt(n+1)))=1`

lim_(n rarr oo)(1+sqrt(n))/(1-sqrt(n))

lim_(n->oo) ((sqrt(n^2+n)-1)/n)^(2sqrt(n^2+n)-1)

lim_(n rarr oo)(sqrt(n+1)-sqrt(n))=0

lim_(n rarr oo)((sqrt(n+3)-sqrt(n+2))/(sqrt(n+2)-sqrt(n+1)))

let f(x)=lim_(n rarr oo)(x^(2n)-1)/(x^(2n)+1)

If f is continuous and differentiable function and f(0)=1,f(1)=2, then prove that there exists at least one c in[0,1] for which f'(c)(f(c))^(n-1)>sqrt(2^(n-1)), where n in N.

If f is continuous and differentiable function and f(0)=1,f(1)=2, then prove that there exists at least one c in [0,1]for which f^(prime)(c)(f(c))^(n-1)>sqrt(2^(n-1)) , where n in Ndot

If f is continuous and differentiable function and f(0)=1,f(1)=2, then prove that there exists at least one c in [0,1]forw h i c hf^(prime)(c)(f(c))^(n-1)>sqrt(2^(n-1)) , where n in Ndot

If f is continuous and differentiable function and f(0)=1,f(1)=2, then prove that there exists at least one c in [0,1]forw h i c hf^(prime)(c)(f(c))^(n-1)>sqrt(2^(n-1)) , where n in Ndot