Home›Calculus 1›Limits by direct substitution›Problem 1.5

Limit of \( \displaystyle \frac{x + 1}{x - 3} \) as \( x \to 1 \)

Problem 1.5 · easy Video

Evaluate \( \displaystyle \lim_{x \to 1} \frac{x + 1}{x - 3} \).

\[ \lim_{x \to 1^+}\left(\frac{x + 1}{x - 3}\right) \]
limitStart with the limit of the function as x approaches 1.✓ Proved
\[ = -1 \]
substitution algebraSubstitute the value x = 1 into the expression. Simplify the numerator and denominator.✓ Proved
\[ = -1.0 \]
limitEvaluate the final numerical value.✓ Proved

Answer \( -1 \)

The graph of f(x) = (x + 1)/(x - 3). Two points slide along the curve toward x = 1, 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 - 3 = 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: pass
qwen3.6:27b-mlx: pass — The solution correctly applies direct substitution for a continuous function at the given point. Each step changes only one aspect of the expression and uses appropriate labels from the allowed vocabulary.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-03 — The solution correctly applies direct substitution for a continuous function at the given point. Each step changes only one aspect of the expression and uses appropriate labels from the allowed 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 — The solution correctly applies direct substitution to evaluate the limit of a rational function at a point where it is continuous. Each step changes only one thing and uses valid labels.
gpt-oss:20b: fail (style) 2026-09-29 — Step 4 is labeled "limit" but it merely evaluates the numeric result; the correct label would be "simplify" or "evaluate". No mathematical errors otherwise.

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.