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Numerade Educator

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Problem 74 Hard Difficulty

Prove that
$$ \displaystyle \lim_{x\to \infty} \frac{\ln x}{x^p} = 0 $$
for any number $ p > 0 $. This shows that the logarithmic function approaches infinity more slowly than any power of $ x $.

Answer

$\lim _{x \rightarrow \infty} \frac{1}{p x^{p}}=\frac{1}{\infty}=0$

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Video Transcript

Okay, we know we have indeterminant form insanity over infinity, implying we have to use law. Attwell's rule. Okay, let's take the derivative of the top derivative of natural Have access simply one of racks. Derivative of extra p is P which is the coefficient of the experiment p x to the P minus one. You have the limit as X approaches infinity off one over p x, the p. It's the same thing as one over infinity, which is simply zero.