garnirativ8

2022-10-03

lHopitals $\underset{x\to \mathrm{\infty}}{lim}\phantom{\rule{thickmathspace}{0ex}}(\mathrm{ln}x{)}^{3x}$?

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: $\mathrm{ln}{x}^{\frac{1}{3x}}$ then I'd just simply rewritten it as $\frac{1}{3x}\cdot \mathrm{ln}x=\frac{\mathrm{ln}x}{3x}$

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: $\mathrm{ln}{x}^{\frac{1}{3x}}$ then I'd just simply rewritten it as $\frac{1}{3x}\cdot \mathrm{ln}x=\frac{\mathrm{ln}x}{3x}$

beshrewd6g

Beginner2022-10-04Added 12 answers

Why do you hant to use l'hopital ?

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

and since $3x\mathrm{ln}(\mathrm{ln}(x))\underset{x\to \mathrm{\infty}}{\u27f6}\mathrm{\infty}$,

$$\underset{x\to \mathrm{\infty}}{lim}(\mathrm{ln}x{)}^{3x}=\mathrm{\infty}.$$

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

and since $3x\mathrm{ln}(\mathrm{ln}(x))\underset{x\to \mathrm{\infty}}{\u27f6}\mathrm{\infty}$,

$$\underset{x\to \mathrm{\infty}}{lim}(\mathrm{ln}x{)}^{3x}=\mathrm{\infty}.$$

ter3k4w8x

Beginner2022-10-05Added 4 answers

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

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

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

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

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

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