It is well known that for a function that is integrable on [0,∞ ), its limit at infinity may not exist. First we illustrated this statement with an example. Then, we present conditions that guarantee the existence of the limit in the following two cases: When the integrable function is non-negative, if the first, second, third, or fourth, derivative is bounded in a neighborhood of each local maximum of f, then the limit exists. Without the non-negative constraint, if an integrable function has a bounded derivative on the entire interval [0,∞ ), then the limit exists.
Prof. Julio G. Dix, Professor of Mathematics, Texas State University, San Marcos Texas
Dix, James Patrick
"Existence of the limit at infinity for a function that is integrable on the half line,"
Rose-Hulman Undergraduate Mathematics Journal: Vol. 14
, Article 1.
Available at: http://scholar.rose-hulman.edu/rhumj/vol14/iss1/1