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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.$\lim _{x \rightarrow \infty} x \sin (\pi / x)$

$\lim _{x \rightarrow \infty} x \sin \frac{\pi}{x}=\pi$

Calculus 1 / AB

Chapter 4

Applications of Derivatives

Section 3

L'Hospital's Rule: Comparing Rates of Growth

Differentiation

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Okay, so here we have the limit as X approaches infinity of x times sign of Pyrex. And so we're gonna rewrite this so that we can use low Beatles rules. And so this is going to be essentially the same thinking as the limit as X approaches infinity of high times. Sign of pi over Rex over. Pi over Rex, Right? And so what happens is these two pies would cancel out, and so this is the same thing. But now when we evaluate at infinity, we get pi times thes sign of pi over infinity over pi over infinity. It's That's gonna be, uh, anything over infinity. We treat like a zero. Okay, so you have pi times sign a zero and sign of 00 And then this pilot a penny become zero and that that is in the form of low p tall. And so the crack into terminate form s O. We can go ahead and use low Beatles rule. And so that means that we could take the derivative of the top and the bottom. So this is the same as the limit as X approaches infinity of Yes, we take the derivative of pi times Sign of hire Extra pies Just a constant suits Pi times the derivative of science Have Rex that's gonna become co sign. Uh, hi over x times. Negative pie over X squared. Okay. And then when we take the derivative of pi over X here, we're gonna end up with negative pi over x squared when that we have a common factor. So we we can cancel out this native pi over X squared, negative, high over X squared and just concern ourselves with the limit as X approaches Infinity of pie, a Times co sign pi over X. Okay, So when we evaluate that high times CO sign of high over X Ah, But at infinity so high over infinity, that's gonna be the same as pie A Times Co sign zero and co sign a zero equal one. So this is just equal to pi and then that is our solution

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