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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 $f$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
Analysis
Hard
##### Suggested Book
Real Analysis; Bartle and Sherbert

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.

# Connected Program at Cheenta

#### 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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