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

1 Answer
May 21, 2016

#0#

Explanation:

This is in the indeterminate form #oo/oo#, so we can apply l'Hôpital's rule, which states that we can take the derivative of the numerator and denominator and then plug in #oo# again to find the limit. Therefore

#lim_(nrarroo)ln(n)/n=lim_(nrarroo)(1/n)/1=lim_(nrarroo)1/n=1/oo=0#

We can also analyze this intuitively: the linear function #n# rises at a greater rate than the logarithmic function #ln(n)#, so since the function that rises faster is in the denominator, the function will approach #0#.

If the function had been flipped, we'd see that the limit as #n# approaches #oo# in #n/ln(n)# would be #oo#.