Limit of \( \displaystyle \left(x + 1\right)^{2} e^{- x - 1} \) as \( x \to \infty \)

Problem 1.114 · medium

1. \[ \lim_{x \to \infty}\left(\left(x + 1\right)^{2} e^{- x - 1}\right) \]
   limit rewriteEvaluate the limit of the function as x approaches infinity. Rewrite the negative exponent using the property exp(-a) = 1/exp(a).✓ Proved
2. \[ = \lim_{x \to \infty}\left(\left(x^{2} + 2 x + 1\right) e^{- x - 1}\right) \]
   algebraExpand the numerator.✓ Proved
3. \[ = \lim_{x \to \infty}\left(\frac{\frac{d}{d x} \left(x^{2} + 2 x + 1\right)}{\frac{d}{d x} e^{x + 1}}\right) \]
   lhopitalApply L'Hopital's Rule because the limit is an indeterminate form of type infinity/infinity.✓ Proved
4. \[ = \lim_{x \to \infty}\left(\left(2 x + 2\right) e^{- x - 1}\right) \]
   simplifyCompute the derivatives of the numerator and denominator.✓ Proved
5. \[ = \lim_{x \to \infty}\left(\frac{\frac{d}{d x} \left(2 x + 2\right)}{\frac{d}{d x} e^{x + 1}}\right) \]
   lhopitalApply L'Hopital's Rule again.✓ Proved
6. \[ = \lim_{x \to \infty}\left(2 e^{- x - 1}\right) \]
   simplifyCompute the derivatives of the numerator and denominator.✓ Proved
7. \[ = \lim_{x \to \infty} 0 \]
   infinityAs x approaches infinity, exp(x + 1) approaches infinity.✓ Proved
8. \[ = 0 \]
   limitThe limit is 0.✓ Proved