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Numerade Educator



Problem 81 Hard Difficulty

Suppose $ f $ is continuous on $ [0, \infty) $ and $ \lim_{x\to\infty} f(x) = 1 $. Is it possible that $ \displaystyle \int_0^\infty f(x)\ dx $ is convergent?




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Video Transcript

the problem is, suppose is continuous on interval from territory Infinity on the limit. My fax my ex goes to infinity is they want It seemed impossible that any girl from zero to infinity about axe is converted. The other is no. The reason is that's the definition of continuous function on bits of probably that affect the limits of our eyes. On Mexico's to invent is one. We have the following results. There must be a good list. Some number, eh? On some number I'm It's through this war on their own. Small one one x is greatest on, eh? The value of a fax is greatest on. Um no, this's a definition of continuous function on DH. It's the definition of limited backs. Goes to independently of our fax you one Now, Auntie girl from zero to infinity of fax The axe s s legal. Choose an integral from zero to a max. The axe us et tu impunity of fax the ex No. And this card it's a fax is continuous. This is a definite integral. There will be some constant number. It is a constant number on DH here, but this integral from eighty Infinity base one axe is correct and the creators are a fax is criticism. Um, So this part is readers up into girl from a to infinity, some on the X and this and stammered and yeah, so our function a fax from zero to infinity is also that gradient.