Question

How do you find the limit of `ln(2x)-ln(1+x)` as x approaches infinity?

Answer 1

Use the property that the difference to two logarithms is division within argument.
Use L'Hôpital's rule on argument.

Explanation:

Given: `lim_(xtooo)(ln(2x) - ln(x + 1))`

Use the property that the difference to two logarithms is division within argument.

`lim_(xtooo)ln((2x)/(x + 1))`

`ln(lim_(xtooo)(2x)/(x + 1))`

Use L'Hôpital's rule

Compute the derivative of the numerator:

`(d(2x))/dx = 2`

Compute the derivative of the denominator:

`(d(x+ 1))/dx = 1`

The new expression is:

`ln(lim_(xtooo)(2/1)) = ln(2)`

Therefore, the original limit is the same:

`lim_(xtooo)(ln(2x) - ln(x + 1)) = ln(2)`