If `f(x) = 1/(sqrt(x))`, evaluate `underset(hrarr0)(Lt)(f(x+h)-f(x))/(h)`.

Let f: R rarr R be a positive increasing function with underset(xrarrinfty)lim(f(3x))/(f(x))=1, then underset(xrarrinfty)lim(f(2x))/(f(x)) =

If f (x) = sqrtx , prove that : (f(x+h)-f(x))/h=1/(sqrt(x+h)+ sqrtx) .

Let the derivative of f(x) be defined as D^(**)f(x)=lim_(hrarr0)(f^(2)(x+h)-f^(2)(x))/(h), where f^(2)(x)={f(x)}^(2) . If u=f(x),v=g(x) , then the value of D^(**)(u.v) is

If f(x) = log ((1-x)/(1+x)) -1 lt x lt 1 then show that f(-x) = -f (x)

For each of the following paris of function f(x) and g(x), verify that: underset(0)overset(1)int[f(x)+g(x)]dx = underset(0)overset(1)intf(x)dx+underset(0)overset(1)intg(x)dx: f(x) = 1, g (x) = x^2

For each of the following paris of function f(x) and g(x), verify that: underset(0)overset(1)int[f(x)+g(x)]dx = underset(0)overset(1)intf(x)dx+underset(0)overset(1)intg(x)dx: f(x) = e^x,g(x) = 1

f(x)=sqrt(9-x^(2)) . find range of f(x).

True or False : If underset(x rarr a)Lt f(x)g(x) exists then both underset(xrarra^+)Lt f(X) and underset(xrarra^-) Lt g(x) exist separately.