The expression n^(c/n) approaches 1 as n approaches infinity for any positive constant c. This can be demonstrated by taking the logarithm of the expression and applying l'Hôpital's rule to evaluate the limit. The logarithmic transformation simplifies the analysis, revealing that the limit converges to zero, leading to the conclusion that the original expression approaches 1.
PREREQUISITESStudents of calculus, mathematicians, and anyone interested in understanding limits and exponential behavior in mathematical analysis.
tolove said:My book is showing this as an intuitive step, but I'm not quite seeing the reasoning behind it.
n**(c/n) → 1 as n → ∞, for, I think, any positive c. But why?
Dick said:Take the log and look at the limit of that. Use l'Hopital's rule, if it doesn't seem intuitive.