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Numerade Educator

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Problem 48 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} x^{3/2} \sin(1/x) $

Answer

$+\infty$

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Video Transcript

Yeah, here we're trying to find the limit of the function as X goes to infinity. And the function were given as X. To the three halves. Um and sign Of one over X. Yeah. So based on this we can let t equal one over X. So this will be T. And then that means actual equal one over T. So then we can rewrite this now as sign of T Over T. to the 3/2. And the reason why we do this is now because we see that T. Is approaching zero plus Uh zero from the right. So then when uh we let T equals zero, we get 0/0. So for that reason we have an interment form and we need to use local tiles rule. We take the derivative of the numerator. We get cosign T. When we take the derivative of the denominator, we get uh t to the one half. We actually had 2/3 key to one half. Okay. Based on this. Now when we plug in here we get one over. Um this would be, so this is 3/2, Be at 1/3/2. Um And then this is going to infinity. So what we end up getting as a result is uh since we have t to one half, T is going to zero from the right in both directions. What we end up getting is 1/0, but it's not really 1/0 because we're approaching it from the right, so it's one over an increasingly small number, so that's going to end up giving us infinity as the answer.