Understanding Limits of Composed Functions at Infinity

[Rectifier]

The problem
$$ \lim_{x \rightarrow \infty} \frac{(\ln x)^{300}}{x} $$

The attempt
$\lim_{x \rightarrow \infty} (\ln x)^{300} = \infty$ since $\lim_{x \rightarrow \infty} f(x) = A$ and $\lim_{x \rightarrow \infty} g(x) = \infty$ thus $\lim_{x \rightarrow \infty}f(g(x)) = A$.

$f(x) = x^{300}$
$g(x) = \ln x$

$\lim_{x \rightarrow \infty} \frac{1}{x} = 0$

So in the end I get $" 0 \cdot \infty "$. Which is not an acceptable solution.

 

[member 587159]

Do you know l'Hospital's rule?

The problem
$$ \lim_{x \rightarrow \infty} \frac{(\ln x)^{300}}{x} $$

The attempt
$\lim_{x \rightarrow \infty} (\ln x)^{300} = \infty$ since $\lim_{x \rightarrow \infty} f(x) = A$ and $\lim_{x \rightarrow \infty} g(x) = \infty$ thus $\lim_{x \rightarrow \infty}f(g(x)) = A$.

$f(x) = x^{300}$
$g(x) = \ln x$

$\lim_{x \rightarrow \infty} \frac{1}{x} = 0$

So in the end I get $" 0 \cdot \infty "$. Which is not an acceptable solution.

 

[Rectifier]

No, I don't. We are supposed to solve it without it at this point.

 

[micromass]

Write $x=e^y$.

 

[Rectifier]

I solved it by setting $x=t^{300}$ but your approach is even better.