lHopitals $ \lim_{x\rightarrow \infty} \; (\ln x)^{3 x} $?

Question

$ \displaystyle \lim_{x\rightarrow \infty} \; (\ln x)^{3 x} =$ ?

Okay, so what do I do with that power? I need to rewrite the term as fractions. How?

If it was the inner function that's in the power of something: $\ln x^{\frac{1}{3 x}}$ then I'd just simply rewritten it as $\frac{1}{3x} \cdot \ln x = \frac{\ln x}{3x}$

Answer 1

Why do you hant to use l'hopital ?

$$(\ln x)^{3x}=e^{3x\ln(\ln(x))}$$

and since $3x\ln(\ln(x))\underset{x\to \infty }{\longrightarrow }\infty $, $$\lim_{x\to\infty }(\ln x)^{3x}=\infty. $$

Answer 2

Why l'Hospital? $\ln x\to+\infty$ as $x\to+\infty$ and $3x\to+\infty$ as $x\to+\infty$. So, you have a limit of type $$ (+\infty)^{+\infty}=+\infty. $$

Answer 3

That isn't a indeterminate form. $\ln x \to \infty$ as $x \to \infty$. So does $3x$.

$\infty^{\infty}=\infty$

The power law for limits comes into play and we get our limit.