Prove that `ln(1+x) < x` for `x < 0.`

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

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

Prove that 1/3 lt log_(20) 3 lt 1/2 .

Using monotonicity prove that (x)/(1-x^(2)) lt tan^(-1)x lt x for every x ge 0

For x in (0,(pi)/(2)) prove that "sin"^(2) x lt x^(2) lt " tan"^(2) x

... sin ^ (8) x <= sin ^ (6) x <= sin ^ (4) x <= sin ^ (2) x <= 1 also ... cos ^ (8) x <= cos ^ (6) x <= cos ^ (4) x <= cos ^ (2) x <= 1

|log_(3) x| lt 2

Prove that tan^(-1) {(x)/(a + sqrt(a^(2) - x^(2)))} = (1)/(2) sin^(-1).(x)/(a), -a lt x lt a

Prove the inequalities tan x gt x + x^(3) //3 " at" 0 lt x lt pi//2