Home›Calculus 1›Limits by direct substitution›Problem 1.44

Limit of \( \displaystyle \frac{x}{x - 4} \) as \( x \to 2 \)

Problem 1.44 · easy Video

Evaluate \( \displaystyle \lim_{x \to 2} \frac{x}{x - 4} \).

\[ \lim_{x \to 2^+}\left(\frac{x}{x - 4}\right) \]
limitStart with the given limit.✓ Proved
\[ = -1 \]
limitSubstitute the value x = 2 into the expression.✓ Proved

Answer \( -1 \)

The graph of f(x) = x/(x - 4). Two points slide along the curve toward x = 2, 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 - 4 = 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 to evaluate the limit of a rational function at a point where the denominator is non-zero. The steps are granular and the labels are appropriate.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-03 — The solution correctly applies direct substitution to evaluate the limit of a rational function at a point where the denominator is non-zero. The steps are granular and the labels are appropriate.
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
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.