Real Analysis Answers

Show that lim 𝑛→∞ 1 /𝑛 [sin( 𝜋/ 𝑛 ) + sin( 2𝜋/ 𝑛 ) + … + sin( 𝑛𝜋/ 𝑛 )] = 2/ 𝜋 .

Show that 𝑈(−𝑓, 𝑝) = −𝐿(𝑓, 𝑝) and 𝐿(−𝑓, 𝑝) = −𝑈(𝑓, 𝑝). 

Let 𝑓 be differentiable. Show that if lim𝑥→∞ 𝑓(𝑥) = 𝐿 ∈ ℝ then lim𝑥→∞ 𝑓 ′ (𝑥) = 0. Provided that the latter limit is existing. Give an example where the converse is not true. Also give an example for which the limit of 𝑓 ′ is not existing even though the limit of 𝑓 is the same as given.

 Let 𝑓 be differentiable. Show that if lim 𝑥→∞

𝑓(𝑥) = 𝐿 ∈ ℝ then lim 𝑥→∞

𝑓′(𝑥) = 0. Provided that the latter limit is existing. Give an example where the converse is not true. Also give an example for which the limit of 𝑓′ is not existing even though the limit of 𝑓 is the same as given.