Understand the problem

Let f be a nonnegative continuous function on \Bbb R s.t \int_0^{\infty}f(t)dt is 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 x goes 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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College Mathematics Program

The higher mathematics program caters to advanced college and university students. It is useful for I.S.I. M.Math Entrance, GRE Math Subject Test, TIFR Ph.D. Entrance, I.I.T. JAM. The program is problem driven. We work with candidates who have a deep love for mathematics. This program is also useful for adults continuing who wish to rediscover the world of mathematics.

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