Question

How do you find the Limit of `(x - ln x)` as x approaches infinity?

Answer 1

combine terms and use L'Hopital's Rule is one way

Explanation:

`x - ln(x)`
`=ln (e^x) + ln(x^{-1})`
`= ln(e^x/x)`

Already it should be clear that this is going to `infty` as the exponential is of greater order.

To be clear we are now actually looking inside the log at `z = lim_{x \to \infty} e^x /x`

and the cheapest shot is using L'Hopital's rule as this `\infty / \infty` form is indeterminate. By L'Hopital's Rule:

`lim_{x \to \infty} e^x /x = lim_{x \to \infty} e^x /1 \to \infty`

and so as `z \to infty`, `ln z \to infty`