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College Mathematics

Integration of nonnegative function: TIFR 2018 Part A, Problem 4

This problem is a cute and simple application of the integration of nonnegative function in the analysis section. It appeared in TIFR GS 2018.

Understand the problem

Let fbe a nonnegative continuous function on \Bbb Rs.t \int_0^{\infty}f(t)dtis finite then \lim_{x \to \infty}f(x)=0
Source of the problem
TIFR 2018 Part A, Problem 4
Topic
Analysis
Difficulty Level
Hard
Suggested Book
Real Analysis; Bartle and Sherbert

Start with hints

Do you really need a hint? Try it first!
Can you think of a function such that it has finite area but as xgoes to infinity, f(x)goes to infinity?

Try to sketch the above function. Picture

See if the lim f(x)exists then it must be zero right? but what if the limit fails to exist and the integral still gives you finite value? Hence, this is the case drawn in the above picture. Hence the answer is false generally.

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