Home›Calculus 1›Limits by direct substitution›Problem 1.24

Limit of \( \displaystyle \frac{\cos{\left(x \right)}}{x + 1} \) as \( x \to 0 \)

Problem 1.24 · easy Video

Evaluate \( \displaystyle \lim_{x \to 0} \frac{\cos{\left(x \right)}}{x + 1} \).

\[ \lim_{x \to 0^+}\left(\frac{\cos{\left(x \right)}}{x + 1}\right) \]
limitStart with the original limit expression.✓ Proved
\[ = 1 \]
limit simplify simplifySince the denominator is non-zero, we can evaluate the limit by direct substitution. Evaluate the numerator and denominator. Final result.✓ Proved

Answer \( 1 \)

The graph of f(x) = cos(x)/(x + 1). 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 + 1 = 0
3	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0
4	✓ 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: fail (style) — Step 2 applies the limit‑law for quotients (since the denominator’s limit is non‑zero) but labels it simply as "limit". The correct label from the allowed vocabulary is "limit-law". No algebraic error occurs, but the labeling is inaccurate.
qwen3.6:27b-mlx: pass

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-03
gpt-oss:20b: fail (style) 2026-10-03 — Step 2 applies the limit‑law for quotients (since the denominator’s limit is non‑zero) but labels it simply as "limit". The correct label from the allowed vocabulary is "limit-law". No algebraic error occurs, but the labeling is inaccurate.
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 direct substitution for a continuous function at a point where the denominator is non-zero. Each step changes only one thing and uses valid labels.
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.