Real Analysis Answers

examine the f:r->r defined by f(x)={1/6(x+1)^3 x is not equal to 0 5/6 x=0} for continuity on R.If it is not continuous at any point of R,find the nature of discontinuity there

lim [(x-3)/(x-1)]^x = 1/e^2

x→∞

Suppose that 𝑓 is differentiable and 𝑓′′(𝑎) exists. Prove that 𝑓′′(𝑎)=lim ℎ→𝑎[(𝑓(𝑎+ℎ)−2𝑓(𝑎)+𝑓(𝑎−ℎ))/h^2.] Give an example where the above limit exists, but 𝑓′′(𝑎) does not exist.

For the function f(x)= x^2-2 defined over[1,5],

Verify L(P,f)≤U(P,f) where is the partition which divide [1,5] into four equal intervals

Let {an} sequence is defined as a1=3, an+1=1/5(an)

converges to zero

Suppose that 𝑓 is differentiable and 𝑓′′(𝑎) exists. Prove that 𝑓′′(𝑎)=limℎ→𝑎𝑓(𝑎+ℎ)−2𝑓(𝑎)+𝑓(𝑎−ℎ)ℎ2. Give an example where the above limit exists, but 𝑓′′(𝑎) does not exist.

-2 is the limit point of [-3,2]

True or false with full explanation

Examine the convergence of the series

a) 3×4/5^2+5×6/7^2+7×8/9^2+.........

b) 1+4x+4^2x^2+4^3x^3+......(x>0)

Let f:[-3,3 ]->R defined by f(x)=5(x)+x^3,where [x]denotes the greatest integer < =x.show that this function is integrable