Real Analysis Answers

Show that the function f given by
f(x)=1/[x+2]^3 ∀x∈]-2,2[
is continuous but not bounded in ]− 2, 2 [

Evaluate the following limit if it exists:
limx→0 [4x^3]/[tan^(3) x+tanx-x]

Check whether the following functions are continuous or not at x = 0. Also, find
the nature of discontinuity at that point, if it exists.
(i) f(x)= {[(2-x)^(1/2) - (2+x)^(1/2)] / [x] , x≠0 ; 1/√2 , x=0
(ii) f(x)= { x^(2)+1/3, x≤0 ; -[x^(3)+1/3], x>0

If a ∈ R is such that o≤a≤ε ∀ ε > 0, then show that a = 0.

The function f defined by = |x-5/2| ,x ∈ R has a local maxima of x=5/2. true/false? justify

A Riemann integrable function is not necessarily differentiable.true/false? justify

Every function differentiable on [a, b] is bounded on [a, b]. true/false ? justify

1/2 is a limit of the interval ]-2.5,1.5[. true/false justify

Q. If f, g is Reimann integrable on [a, b] then fg is reimann integrable on [a, b].

Price √2 is irrational