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

Problem 1.424 · medium

1. \[ \lim_{x \to \infty}\left(\left(x - 1\right)^{2} e^{1 - x}\right) \]
   limit rewriteStart with the original limit. Rewrite the exponential term using negative exponents.✓ Proved
2. \[ = \lim_{x \to \infty}\left(\left(x^{2} - 2 x + 1\right) e^{1 - x}\right) \]
   algebraExpand the squared binomial.✓ 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 since the limit is of the form infinity/infinity.✓ Proved
4. \[ = \lim_{x \to \infty}\left(\left(2 x - 2\right) e^{1 - x}\right) \]
   simplifyCompute the derivatives.✓ 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. \[ = \lim_{x \to \infty}\left(2 e^{1 - x}\right) \]
   simplifyCompute the derivatives.✓ Proved
7. \[ = \lim_{x \to \infty} 0 \]
   infinityEvaluate the limit as x approaches infinity.✓ Proved
8. \[ = 0 \]
   limitThe final value of the limit.✓ Proved