Limit of \( \displaystyle 9 x^{2} e^{- 3 x} \) as \( x \to \infty \)

Problem 1.169 · medium

1. \[ \lim_{x \to \infty}\left(9 x^{2} e^{- 3 x}\right) \]
   limit rewriteIdentify the limit to be evaluated. Rewrite the exponential with a negative exponent as a fraction.✓ Proved
2. \[ = \lim_{x \to \infty}\left(\frac{\frac{d}{d x} 9 x^{2}}{\frac{d}{d x} e^{3 x}}\right) \]
   lhopitalApply L'Hopital's Rule because the limit is of the form infinity/infinity.✓ Proved
3. \[ = \lim_{x \to \infty}\left(6 x e^{- 3 x}\right) \]
   simplify simplifyCompute the derivatives of the numerator and denominator. Simplify the resulting fraction.✓ Proved
4. \[ = \lim_{x \to \infty}\left(\frac{\frac{d}{d x} 6 x}{\frac{d}{d x} e^{3 x}}\right) \]
   lhopitalApply L'Hopital's Rule again because the limit is still of the form infinity/infinity.✓ Proved
5. \[ = \lim_{x \to \infty}\left(2 e^{- 3 x}\right) \]
   simplify simplifyCompute the derivatives of the numerator and denominator. Simplify the resulting fraction.✓ Proved
6. \[ = \lim_{x \to \infty} 0 \]
   infinityAs x approaches infinity, the denominator grows without bound.✓ Proved
7. \[ = 0 \]
   limitThe limit evaluates to zero.✓ Proved