Limits - composed functions

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1. Jul 15, 2016

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.

2. Jul 15, 2016

Math_QED

Do you know l'Hospital's rule?

3. Jul 15, 2016

Rectifier

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

4. Jul 16, 2016

micromass

Staff Emeritus
Write $x=e^y$.

5. Jul 16, 2016

Rectifier

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