Limit of \( \displaystyle 4 x^{2} e^{- 2 x} \) as \( x \to \infty \)

Problem 1.110 · medium

1. \[ \lim_{x \to \infty}\left(4 x^{2} e^{- 2 x}\right) \]
   limit rewriteEvaluate the limit of the function as x approaches infinity. Rewrite the negative exponent as a fraction.✓ Proved
2. \[ = \lim_{x \to \infty}\left(\frac{\frac{d}{d x} 4 x^{2}}{\frac{d}{d x} e^{2 x}}\right) \]
   lhopitalApply L'Hopital's rule because the limit is an indeterminate form of type infinity/infinity.✓ Proved
3. \[ = \lim_{x \to \infty}\left(4 x e^{- 2 x}\right) \]
   simplify simplifyCompute the derivatives in the numerator and denominator. Simplify the fraction by canceling the common factor of 2.✓ Proved
4. \[ = \lim_{x \to \infty}\left(\frac{\frac{d}{d x} 4 x}{\frac{d}{d x} e^{2 x}}\right) \]
   lhopitalApply L'Hopital's rule again.✓ Proved
5. \[ = \lim_{x \to \infty}\left(2 e^{- 2 x}\right) \]
   simplify simplify rewriteCompute the derivatives in the numerator and denominator. Simplify the fraction. Rewrite the expression using a positive exponent.✓ Proved
6. \[ = 0 \]
   limitThe limit of exp(-2x) as x approaches infinity is 0.✓ Proved