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.

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