Show that `x/((1+x))<1n(1+x)` for `x>0`

Prove that (x)/((1 + x)) lt log (1 + x) lt x " for " x gt 0

Show that f(x)=(x)/(sqrt(1+x))- ln (1+x) is an increasing function for x gt -1 .

If x gt -1 , show that : (x)/(sqrt(1+x))-log(1+x)+9 is an increasing function of x.

Using monotonicity prove that x lt - ln (1-x) lt x (1-x)^(-1) for 0 lt x lt 1

Let g^(prime)(x)>0a n df^(prime)(x)<0AAx in Rdot Then (f(x+1))>g(f(x-1)) f(g(x-1))>f(g(x+1)) g(f(x+1))

Prove that ln (1+1/x) gt (1)/(1+x), x gt 0 . Hence, show that the function f(x)=(1+1/x)^(x) strictly increases in (0, oo) .

Statement-1 : The function f defined as f(x) = a^(x) satisfies the inequality f(x_(1)) lt f(x_(2)) for x_(1) gt x_(2) when 0 lt a lt 1 . and Statement-2 : The function f defined as f(x) = a^(x) satisfies the inequality f(x_(1)) lt f(x_(2)) for x_(1) lt x_(2) when a gt 1 .

If f(x) is defined as follows: f(x){{:(1,x,gt0),(-1,x,lt0),(0,x,=0):} Then show that lim_(xrarr0) f(x) does not exist.

If f(x) is continuous at x=0 , where f(x)={((sin x)/(x)+cos x", for " x gt 0),((4(1-sqrt(1-x)))/(x)", for " x lt 0):} , then f(0)=