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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.$\lim _{x \rightarrow \infty} x^{3} e^{-x^{2}}$

$\lim _{x \rightarrow \infty} \frac{3}{4 x e^{x^{2}}}=\frac{3}{4(\infty) e^{(\infty)^{2}}}=\frac{3}{\infty}=0$

Calculus 1 / AB

Chapter 4

Applications of Derivatives

Section 3

L'Hospital's Rule: Comparing Rates of Growth

Differentiation

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do we need to solve the limit? As X approaches infinity of X cubed times e to the power of native X squared. And so each of the power of negative X squared is really one divided by Ito power of X squared. So when we rewrite it in that form, so it would be the limit as X approaches infinity of execute over e to the power of X squared, Then we can he violate this limit. So let's evaluate at infinity and we get infinity cubed over e to the power of infinity squared, which is, of course, infinity over infinity. So this means we can use Blue Beetle's rule here and take the derivative. So let's take the derivative, uh, both the numerator and denominator. So the derivative of the numerator being three x squared and the derivative of the dominator being two x times eat the power of X squared. You know, we actually get a cancellation, right? We can get rid of one of these exes us. This is going to be the same. Oust the limit as X approaches infinity of three x over two times e to the power of X squared. So we will evaluate again and we'll get three times infinity over two times E to the power of infinity squared, which is infinity over infinity again. A solely used Loki tells Rule one more time or who knows how many more times, but at least one more time. And so that will give us the limit as exit, purchase and Fanny. Now you want the derivative of three X, which is three and then the derivative of two times each, the power of ex squares that's gonna be It's gonna be two times, two x times each part of X squared, which will be four times E to the power of X squared. Okay, so now we will evaluate the limit again, and in this case, we're gonna get three over four times E to the power of X squared. But not an X ray. We're evaluating it. So it's gonna be infinity squared, which is equivalent to three over infinity, uh, which is gonna be equal to zero. So the limit is equal to

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