If `f(x) = 3x^7`, evalaute `underset(trarr0)Lt(f(1+t)-f(1))/(t)`

If a, b and c are real numbers, then the value of underset(trarr0)(lim)((1)/(t)int_(0)^(t)(1+asinbx)^(c//x)dx) equals

If f is diffenrentiable at x = 1 , then underset (xto1) ("limit") (xf(1)-f(x))/(x-1) is equal to :

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)) =

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

If f(x) is of the form f(x) = a + bx + cx^2 , show that underset(0)overset(1)intf(x) dx = 1/6 [f(0)+4f(1/2)+f(1)]

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.

If a function satisfies the relation f(x) f''(x)-f(x)f'(x)=(f'(x))^(2) AA x in R and f(0)=f'(0)=1, then The value of underset(x to -oo)(lim)f(x) is

True or False: If f(2a-x)=-f(x) , then underset0overset(2a)int f(x)dx=0 .