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Problem 32 Medium Difficulty

Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. If l'Hospital's Rule doesn't apply, explain why.

$ \displaystyle \lim_{x\to \infty} \frac{(\ln x)^2}{x} $

Answer

$\lim _{x \rightarrow \infty} \frac{\ln (x)^{2}}{x}=0$

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Video Transcript

Yeah, Okay, fine. That the limit us locked us rule there a proper great if they're some are elementary. My third Consider using it. Okay, here is the limit as X goes to infinity. So if you're plucking infinity to the cop log rhythm functional algorithm infinity, which is infinity. So you square it, Takis infinity. And he wanted by another run that another x x goes to infinity. So there's this infinite he divided by infinity, which is saying determined the phone so we can use the locked house rule to write it as the limit as the axe goes to infinity that you're a tip off that hop because two times Lok X, use the power rule force first and then use the train rule to find a curative off his side function that isthe one over X. And then he wanted by the curative off the denominator that hiss one. So, actually you just need to find the limit off this function and that his Seiko too. Since the limit has the linear property, you can put a truly in front that Islamic as X goes to infinity now algorithm function of X times where our ex that means this one divided by X. Okay, So here you feel plaque in infinity to here and here. That is also infinite. He divided by infinity, which is? They come in the form so you can use the love house rule again. Practice as the limit. A few times the limit has goes to infinity that the early if off the top fishes where our ex divided by the curative off the denominator like this one. So you get the limit as axe ghost waiting for that hero our acts that his zero so two times zero is zero.