Home›Calculus 1›L'Hôpital's rule›Problem 1.8

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

Problem 1.8 · easy Video

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

\[ \lim_{x \to 0^+}\left(\frac{e^{x} - 1}{x}\right) \]
limitStart with the original limit.✓ Proved
\[ = \lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(e^{x} - 1\right)}{\frac{d}{d x} x}\right) \]
lhopitalApply L'Hopital's rule because the limit is of the form 0/0.✓ Proved
\[ = \lim_{x \to 0^+} e^{x} \]
simplify simplifyDifferentiate the numerator and denominator. Simplify the expression.✓ Proved
\[ = 1 \]
limit simplifyEvaluate the limit by substituting x = 0. Final result.✓ Proved

Answer \( 1 \)

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

✓ 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 undefined where Derivative(x, x) = 0
3	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where Derivative(x, x) = 0
4	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0
5	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0
6	✓ 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: fail (error) — Step 3 is labeled 'simplify' but performs the differentiation required by L'Hôpital's rule; the differentiation should have been part of the 'lhopital' step or explicitly labeled 'derivative'. Applying two distinct logical operations (invoking the rule and performing the differentiation) across separate steps with incorrect labeling violates the single-rule-per-step constraint.

Senior review claude-sonnet-5-5, 2026-10-03: pass — L'Hopital's rule is applied correctly to the 0/0 form. The steps then evaluate the derivatives, drop the /1, substitute x=0, and simplify exp(0) to 1, all of which are correct and labeled with permitted rules. The step 3 note says 'differentiate' under 'simplify', which is slightly loose but not wrong.

qwen3.6:27b-mlx: dismiss — The vocabulary has no 'derivative' label; step 2 only sets up the ratio of unevaluated derivatives, and step 3 evaluates them, which 'simplify' (or 'lhopital') legitimately covers. SymPy proved the line equal to the previous one.

Every verdict on record (7)

qwen3.6:27b-mlx: fail (error) 2026-10-03 — Step 3 is labeled 'simplify' but performs the differentiation required by L'Hôpital's rule; the differentiation should have been part of the 'lhopital' step or explicitly labeled 'derivative'. Applying two distinct logical operations (invoking the rule and performing the differentiation) across separate steps with incorrect labeling violates the single-rule-per-step constraint.
gpt-oss:20b: pass 2026-10-03
claude-sonnet-5-5: pass 2026-10-03 — L'Hopital's rule is applied correctly to the 0/0 form. The steps then evaluate the derivatives, drop the /1, substitute x=0, and simplify exp(0) to 1, all of which are correct and labeled with permitted rules. The step 3 note says 'differentiate' under 'simplify', which is slightly loose but not wrong.
qwen3.6:27b-mlx: fail (error) 2026-09-29 — Step 3 is labeled 'simplify' but performs differentiation, which is not a simplification rule. The differentiation should have been labeled 'derivative' in Step 2 or 3, but Step 2 already claimed 'lhopital' (which implies differentiation). Applying differentiation in Step 3 without a 'derivative' label, or labeling it 'simplify', is a defect because 'simplify' does not cover differentiation.
gpt-oss:20b: pass 2026-09-29
qwen3.6:27b-mlx: fail (error) 2026-09-29 — Step 3 is labeled 'simplify' but performs differentiation, which is not a simplification step. The differentiation should have been labeled 'derivative' in Step 2 or Step 3, but Step 2 already claimed 'lhopital' which implies the differentiation happened there; however, Step 3 explicitly shows the result of differentiation while claiming to just simplify, creating a logical disconnect where the differentiation action is unlabelled or mislabelled.
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.