How to prove that ln ⁡ ( x ) &lt; 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 ) &lt; 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 | | &lt; 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.

Question

How to prove that   ln  ⁡  (  x  )  &amp;lt;  x for   x  →  ∞During my calculus homework I need to prove some limits without using L&#039;Hôpital&#039;s rule. I have difficulties to show rigorously that   ln  ⁡  (  x  )  &amp;lt;  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          |                          |    &amp;lt;      x    4  I&#039;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&#039;t find the way to make it formal enough.

popman14ee · Accepted Answer

For the first part:Note the following:1.  log  ⁡  (  1  )  =  0  &amp;lt;  12.      log    ′    ⁡  (  x  )  =      1    x    ≤  1  =      d          d      x        x for   x  ≥  13.      log    ′    ⁡  (  x  )  =      1    x    ≥  1  =      d          d      x        x for   x  ≤  1So you can integrate over an appropriate interval to get   log  ⁡  (  x  )  &amp;lt;  x for all   x.For the second part:Try to use      lim          x      →      ∞                  e      x              x      n        =  ∞for any natural number   n.

preityk7t · Accepted Answer

Hint: Consider the function   f  (  x  )  =  ln  ⁡  (  x  )  −  x. Note that   f  (  1  )  &amp;lt;  0 and show this function is monotonically decreasing (from   x  =  1) by taking the derivative