Home›Calculus 1›Limits by rationalizing›Problem 1.10

Limit of \( \displaystyle \frac{\sqrt{x + 4} - 2}{x} \) as \( x \to 0 \)

Problem 1.10 · medium Video

Evaluate \( \displaystyle \lim_{x \to 0} \frac{\sqrt{x + 4} - 2}{x} \).

\[ \lim_{x \to 0^+}\left(\frac{\sqrt{x + 4} - 2}{x}\right) \]
limit algebraStart with the limit as x approaches 0 from the right. Multiply the numerator and denominator by the conjugate.✓ Proved
\[ = \lim_{x \to 0^+} \frac{1}{\sqrt{x + 4} + 2} \]
simplify cancel simplifySimplify the numerator. Cancel the common factor of x. Simplify the remaining expression.✓ Proved
\[ = \frac{1}{4} \]
limit simplify simplifyEvaluate the limit by substituting x = 0. Simplify the square root. Final result.✓ Proved

Answer \( \frac{1}{4} \)

The graph of f(x) = (sqrt(x + 4) - 2)/x. Two points slide along the curve toward x = 0, one from each side; their heights close in on 1/4.

✓ 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
3	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where x = 0 undefined where sqrt(x + 4) + 2 = 0
4	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where sqrt(x + 4) + 2 = 0
5	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where sqrt(x + 4) + 2 = 0
6	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where sqrt(x + 4) + 2 = 0
7	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0
8	✓ 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 the conjugate method to resolve the indeterminate form. Each step changes only one aspect of the expression and uses appropriate labels from the fixed vocabulary.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-03 — The solution correctly applies the conjugate method to resolve the indeterminate form. Each step changes only one aspect of the expression and uses appropriate labels from the fixed vocabulary.
gpt-oss:20b: pass 2026-10-03
qwen3.6:27b-mlx: pass 2026-09-29 — The solution correctly applies the conjugate method to resolve the indeterminate form. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.
gpt-oss:20b: pass 2026-09-29
qwen3.6:27b-mlx: pass 2026-09-29 — The solution correctly applies the conjugate method to resolve the indeterminate form. Each step changes only one aspect of the expression and uses appropriate labels from the fixed vocabulary.
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.