Limit of \( \displaystyle \left(3 x + 1\right)^{2} e^{- 3 x - 1} \) as \( x \to \infty \)

Problem 1.429 · medium

1. \[ \lim_{x \to \infty}\left(\left(3 x + 1\right)^{2} e^{- 3 x - 1}\right) \]
   limit rewriteStart with the original limit. Rewrite the negative exponent using division.✓ Proved
2. \[ = \lim_{x \to \infty}\left(\left(9 x^{2} + 6 x + 1\right) e^{- 3 x - 1}\right) \]
   algebraExpand the squared binomial.✓ Proved
3. \[ = \lim_{x \to \infty}\left(\frac{\frac{d}{d x} \left(9 x^{2} + 6 x + 1\right)}{\frac{d}{d x} e^{3 x + 1}}\right) \]
   lhopitalApply L'Hopital's rule because the limit is of the form infinity/infinity.✓ Proved
4. \[ = \lim_{x \to \infty}\left(\frac{\left(18 x + 6\right) e^{- 3 x - 1}}{3}\right) \]
   simplifyCalculate the derivatives.✓ Proved
5. \[ = \lim_{x \to \infty}\left(\frac{\left(18 x + 6\right) e^{- 3 x}}{3 e}\right) \]
   simplifySplit the exponential term.✓ Proved
6. \[ = \lim_{x \to \infty}\left(\frac{\left(18 x + 6\right) e^{- 3 x}}{3}\right) \]
   simplifyRemove the constant factor exp(1).✓ Proved
7. \[ = \lim_{x \to \infty}\left(\left(6 x + 2\right) e^{- 3 x}\right) \]
   simplifyDivide numerator and denominator by 3.✓ Proved
8. \[ = \lim_{x \to \infty}\left(\frac{\frac{d}{d x} \left(6 x + 2\right)}{\frac{d}{d x} e^{3 x}}\right) \]
   lhopitalApply L'Hopital's rule again.✓ Proved
9. \[ = \lim_{x \to \infty}\left(2 e^{- 3 x}\right) \]
   simplify simplifyCalculate the derivatives. Simplify the fraction.✓ Proved
10. \[ = \lim_{x \to \infty} 0 \]
    infinityAs x approaches infinity, the denominator grows to infinity.✓ Proved
11. \[ = 0 \]
    limitThe limit evaluates to 0.✓ Proved