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 $\frac{1}{3x} \cdot \ln x = \frac{\ln x}{3x}$

$\infty$ is right – Dr. Sonnhard Graubner Oct 02 '15 at 18:50 — Dr. Sonnhard Graubner, Oct 02 '15 at 18:50

score 4 · Accepted Answer · answered Oct 02 '15 at 18:37

4

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

answered Oct 02 '15 at 18:37

Surb

55,662

score 3 · Answer 2 · answered Oct 02 '15 at 18:44

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.

answered Oct 02 '15 at 18:44

Aleksandar

1,771

score 2 · Answer 3 · answered Oct 02 '15 at 18:36

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 $$ (+\infty)^{+\infty}=+\infty. $$

answered Oct 02 '15 at 18:36

mickep

19,962

Why the down vote, if I may ask? – mickep Oct 02 '15 at 19:11
1

I'll neutralize that nefarious down vote! I simply don't understand why some users are so trigger happy to down vote ... especially when the answer is correct. – Mark Viola Oct 02 '15 at 23:14

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

3 Answers3

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