Home›Calculus 1›L'Hôpital's rule›Problem 1.28

Limit of \( \displaystyle \frac{- x + e^{x} - 1}{x^{2}} \) as \( x \to 0 \)

Problem 1.28 · medium Video

Evaluate \( \displaystyle \lim_{x \to 0} \frac{- x + e^{x} - 1}{x^{2}} \).

\[ \lim_{x \to 0^+}\left(\frac{- x + e^{x} - 1}{x^{2}}\right) \]
limit algebraStart with the limit of the given function as x approaches 0 from the right. Rearrange the terms in the numerator.✓ Proved
\[ = \lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(- x + e^{x} - 1\right)}{\frac{d}{d x} x^{2}}\right) \]
lhopitalApply L'Hopital's rule since the limit is an indeterminate form 0/0.✓ Proved
\[ = \lim_{x \to 0^+}\left(\frac{e^{x} - 1}{2 x}\right) \]
simplifyCompute the derivatives in the numerator and denominator.✓ Proved
\[ = \lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(e^{x} - 1\right)}{\frac{d}{d x} 2 x}\right) \]
lhopitalApply L'Hopital's rule again as the limit is still 0/0.✓ Proved
\[ = \lim_{x \to 0^+}\left(\frac{e^{x}}{2}\right) \]
simplifyCompute the derivatives.✓ Proved
\[ = \frac{1}{2} \]
limitEvaluate the limit as x approaches 0.✓ Proved

Answer \( \frac{1}{2} \)

The graph of f(x) = (-x + e^(x) - 1)/x². Two points slide along the curve toward x = 0, one from each side; their heights close in on 1/2.

✓ 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 undefined where x = 0
3	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where x = 0 undefined where Derivative(x**2, x) = 0
4	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where Derivative(x**2, x) = 0 undefined where x = 0
5	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where x = 0 undefined where Derivative(2*x, x) = 0
6	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where Derivative(2*x, x) = 0
7	✓ 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

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-03
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 — The solution correctly applies L'Hopital's rule twice to resolve the 0/0 indeterminate form. Each step changes only one aspect of the expression (either applying the rule or simplifying the resulting derivatives), and the labels are appropriate.
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.