Question

How do you find the Limit of `ln(x+1)/x` as x approaches infinity?

Answer 1

`lim_(x->∞) ln(x+1)/(x) = 0`

Explanation:

This limit is indeterminate because direct substitution yields `∞/∞`.
Therefore, we can apply L'Hospital's rule, which basically is taking a derivative of the numerator and the denominator at the same time.

`lim_(x->∞) ln(x+1)/(x) -> ∞/∞`

Applying L'Hospital's rule gives us

`lim_(x->∞) 1/(x+1) = 0`

This makes sense because if we look at the graph of our original function, we can see that as `x` approaches infinity, it tends to go to zero.

graph{ln(x+1)/(x) [-10, 10, -5, 5]}