Home›Calculus 1›L'Hôpital's rule›Problem 1.415

Limit of \( \displaystyle \frac{\cos{\left(x + 1 \right)} - 1}{e^{x + 1} - 1} \) as \( x \to -1 \)

Problem 1.415 · medium

Evaluate \( \displaystyle \lim_{x \to -1} \frac{\cos{\left(x + 1 \right)} - 1}{e^{x + 1} - 1} \).

\[ \lim_{x \to -1^+}\left(\frac{\cos{\left(x + 1 \right)} - 1}{e^{x + 1} - 1}\right) \]
limitStart with the initial limit expression.✓ Proved
\[ = \lim_{x \to -1^+}\left(\frac{\frac{d}{d x} \left(\cos{\left(x + 1 \right)} - 1\right)}{\frac{d}{d x} \left(e^{x + 1} - 1\right)}\right) \]
lhopitalApply L'Hopital's rule since the limit is an indeterminate form of type 0/0.✓ Proved
\[ = \lim_{x \to -1^+}\left(- e^{- x - 1} \sin{\left(x + 1 \right)}\right) \]
simplifyCompute the derivatives of the numerator and denominator.✓ Proved
\[ = \lim_{x \to -1^+} 0 \]
algebra simplifySubstitute the value x = -1 into the expression. Evaluate the trigonometric and exponential functions.✓ Proved
\[ = 0 \]
simplifyThe final value of the limit is 0.✓ Proved

Answer \( 0 \)

✓ 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.

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 exp(x + 1) - 1 = 0 undefined where Derivative(exp(x + 1) - 1, x) = 0
3	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where Derivative(exp(x + 1) - 1, 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: pass — The solution correctly applies L'Hôpital's rule and simplifies the resulting expression. Each step changes only one aspect of the expression and uses an acceptable label from the fixed vocabulary.

Senior review claude-sonnet-5-5, 2026-10-08: pass — L'Hopital is validly applied to a 0/0 form (cos 0 - 1 = 0 and e^0 - 1 = 0). The derivatives, the evaluation to 0/1, and the final value 0 are all correct. The label on line 4 could be 'limit-law' or 'substitution', but that is only a style point.

qwen3.6:27b-mlx: dismiss — Line 4 is equal to line 3, since the continuous expression at x=-1 has the value -sin(0)/exp(0) and the Limit wrapper is kept, so nothing is false. The step is direct substitution, with x+1 evaluating to 0 as arithmetic, and the 'algebra' label is at worst a loose fit rather than a rule the step did not apply.

Every verdict on record (5)

gpt-oss:20b: pass 2026-10-08
qwen3.6:27b-mlx: pass 2026-10-08 — The solution correctly applies L'Hôpital's rule and simplifies the resulting expression. Each step changes only one aspect of the expression and uses an acceptable label from the fixed vocabulary.
claude-sonnet-5-5: pass 2026-10-08 — L'Hopital is validly applied to a 0/0 form (cos 0 - 1 = 0 and e^0 - 1 = 0). The derivatives, the evaluation to 0/1, and the final value 0 are all correct. The label on line 4 could be 'limit-law' or 'substitution', but that is only a style point.
qwen3.6:27b-mlx: fail (error) 2026-10-08 — Step 4 is labeled 'algebra' but performs a substitution of the limit value into the expression, which should be labeled 'substitution'. Furthermore, Step 4 replaces the variable x with -1 inside the Limit operator, which is logically incorrect; one evaluates the limit by substituting into the simplified expression, not by rewriting the limit expression itself with the value plugged in before the limit is taken.
gpt-oss:20b: pass 2026-10-08

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 2026-10-08 with SymPy 1.14.0.