Home›Calculus 1›L'Hôpital's rule›Problem 1.13

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

Problem 1.13 · easy Video

Evaluate \( \displaystyle \lim_{x \to \infty} x^{2} e^{- x} \).

\[ \lim_{x \to \infty}\left(x^{2} e^{- x}\right) \]
limit rewriteStart with the original limit expression. Rewrite the negative exponent as a fraction.✓ Proved
\[ = \lim_{x \to \infty}\left(\frac{\frac{d}{d x} x^{2}}{\frac{d}{d x} e^{x}}\right) \]
lhopitalApply L'Hopital's Rule because the limit is of the form infinity/infinity.✓ Proved
\[ = \lim_{x \to \infty}\left(2 x e^{- x}\right) \]
simplifyCompute the derivatives of the numerator and denominator.✓ Proved
\[ = \lim_{x \to \infty}\left(\frac{\frac{d}{d x} 2 x}{\frac{d}{d x} e^{x}}\right) \]
lhopitalApply L'Hopital's Rule again because the limit is still of the form infinity/infinity.✓ Proved
\[ = \lim_{x \to \infty}\left(2 e^{- x}\right) \]
simplifyCompute the derivatives of the numerator and denominator.✓ Proved
\[ = \lim_{x \to \infty} 2 \lim_{x \to \infty} e^{- x} \]
limit-lawSplit the limit using the product rule.✓ Proved
\[ = 0 \]
limit algebraEvaluate the limits of the individual components. Final calculation.✓ Proved

Answer \( 0 \)

The graph of f(x) = x²·e^(-x). A point runs right along the curve; its height settles toward 0.

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. How we verify

The full receipt

Line	Status	Checked by	Detail
1	✓ Proved	sympy 1.14.0	line 1 is the problem as stated
2	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0
3	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where Derivative(exp(x), x) = 0
4	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where Derivative(exp(x), x) = 0
5	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where Derivative(exp(x), x) = 0
6	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where Derivative(exp(x), x) = 0
7	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0
8	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0
9	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0
answer	✓ Proved	sympy 1.14.0	final line against the stated answer: simplify(a - b) reduced to 0
answer, a second way	✓ Proved	sympy 1.14.0	SymPy took the limit from both sides and got the stated value

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: pass — The solution correctly applies L'Hopital's Rule twice and uses limit laws to evaluate the final expression. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-03 — The solution correctly applies L'Hopital's Rule twice and uses limit laws to evaluate the final expression. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.
gpt-oss:20b: pass 2026-10-03
qwen3.6:27b-mlx: pass 2026-09-29
gpt-oss:20b: pass 2026-09-29
qwen3.6:27b-mlx: pass 2026-09-29
gpt-oss:20b: pass 2026-09-29

Proved: SymPy reduced the difference between the two sides to zero. Checked independently: a separate method, named above, confirmed it. Checked numerically: the two sides agree at every sampled point, which is evidence, not proof. Reviewed: a model or a person read it; that is all a sentence can have. Solution by gemma4:26b, checked 2026-10-03 with SymPy 1.14.0.