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

1 Answer
Jul 22, 2016

#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]}