Limit of \( \displaystyle x^{2} e^{- x} \) as \( x \to \infty \)

Problem 1.13 · easy

1. \[ \lim_{x \to \infty}\left(x^{2} e^{- x}\right) \]
   limit rewriteStart with the original limit expression. Rewrite the negative exponent as a fraction.✓ Proved
2. \[ = \lim_{x \to \infty}\left(\frac{\frac{d}{d x} x^{2}}{\frac{d}{d x} e^{x}}\right) \]
   lhopitalApply L'Hopital's Rule because the limit is of the form infinity/infinity.✓ Proved
3. \[ = \lim_{x \to \infty}\left(2 x e^{- x}\right) \]
   simplifyCompute the derivatives of the numerator and denominator.✓ Proved
4. \[ = \lim_{x \to \infty}\left(\frac{\frac{d}{d x} 2 x}{\frac{d}{d x} e^{x}}\right) \]
   lhopitalApply L'Hopital's Rule again because the limit is still of the form infinity/infinity.✓ Proved
5. \[ = \lim_{x \to \infty}\left(2 e^{- x}\right) \]
   simplifyCompute the derivatives of the numerator and denominator.✓ Proved
6. \[ = \lim_{x \to \infty} 2 \lim_{x \to \infty} e^{- x} \]
   limit-lawSplit the limit using the product rule.✓ Proved
7. \[ = 0 \]
   limit algebraEvaluate the limits of the individual components. Final calculation.✓ Proved