How to prove that ln ⁡ ( x ) < x for x → ∞During...

Yahir Tucker

Yahir Tucker

Answered

2022-06-26

How to prove that ln ( x ) < x for x
During my calculus homework I need to prove some limits without using L'Hôpital's rule. I have difficulties to show rigorously that ln ( x ) < x for big enough x.
For example, I need to find the image of the continues function f : R R such that for every x R: | f ( x ) x e | x | | < x 4
I've tried to prove that x e | x | x 4 if x , and then the image of f will be all the reals. Unfortunately, I don't find the way to make it formal enough.

Answer & Explanation

popman14ee

popman14ee

Expert

2022-06-27Added 19 answers

For the first part:
Note the following:
1. log ( 1 ) = 0 < 1
2. log ( x ) = 1 x 1 = d d x x for x 1
3. log ( x ) = 1 x 1 = d d x x for x 1
So you can integrate over an appropriate interval to get log ( x ) < x for all x.
For the second part:
Try to use
lim x e x x n =
for any natural number n.
preityk7t

preityk7t

Expert

2022-06-28Added 6 answers

Hint: Consider the function f ( x ) = ln ( x ) x. Note that f ( 1 ) < 0 and show this function is monotonically decreasing (from x = 1) by taking the derivative

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