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

Problem 1.105 · easy

1. \[ \lim_{x \to \infty}\left(2 x e^{- 2 x}\right) \]
   limit rewriteStart with the original limit expression. Rewrite the negative exponent as a fraction.✓ Proved
2. \[ = \lim_{x \to \infty}\left(\frac{\frac{d}{d x} 2 x}{\frac{d}{d x} e^{2 x}}\right) \]
   lhopitalApply L'Hopital's Rule because the limit is of the form infinity/infinity.✓ Proved
3. \[ = \lim_{x \to \infty} e^{- 2 x} \]
   simplify simplify rewriteCompute the derivatives of the numerator and denominator. Simplify the fraction by canceling the common factor of 2. Rewrite the expression with a negative exponent.✓ Proved
4. \[ = 0 \]
   limitEvaluate the limit as x approaches infinity.✓ Proved