Answer to Question #248694 in Real Analysis for user1

Question

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