I am trying to calculate the following limit without L'Hôpital's rule:
$$\lim_{n \to \infty} \dfrac{n}{\ln(n)}$$
I tried every trick I know but nothing works. You don't have to prove it by definition.
Calculus – Limit of n/ln(n) Without L’Hôpital’s Rule
calculusfunctionslimits
Best Answer
Let $\ln(n) = t$. We then have $n = e^t$. Hence, we have $$\lim_{n \to \infty} \dfrac{n}{\ln(n)} = \lim_{t \to \infty} \dfrac{e^t}{t}$$ Recall that $e^t = 1 + t + \dfrac{t^2}{2!} + \cdots > \dfrac{t^2}{2}$. Hence, we have $$\lim_{t \to \infty} \dfrac{e^t}{t} > \lim_{t \to \infty} \dfrac{t^2}{2t} = \lim_{t \to \infty} \dfrac{t}{2} = \infty$$