Home›Calculus 1›Limits by rationalizing›Problem 1.23

Limit of \( \displaystyle \frac{\sqrt{x + 1} - 1}{x} \) as \( x \to 0 \)

Problem 1.23 · medium Video

Evaluate \( \displaystyle \lim_{x \to 0} \frac{\sqrt{x + 1} - 1}{x} \).

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

Answer \( \frac{1}{2} \)

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

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. A reviewer's objection to the explanation stands; see the receipt. 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 + 1) + 1 = 0
4	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where sqrt(x + 1) + 1 = 0
5	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where sqrt(x + 1) + 1 = 0
6	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where sqrt(x + 1) + 1 = 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: fail (error) — Step 2 claims to multiply by the conjugate, but the expression shown multiplies by sqrt(x+1) instead of (sqrt(x+1)+1). This is a mathematical error in the step's content, not just a labeling issue.

Senior review claude-sonnet-5-5, 2026-10-04: fail — The final answer 1/2 is correct and every line is equal to the problem. However, line 2 does not show the conjugate multiplication it claims, and line 3 then does the conjugate multiplication and the numerator simplification in one step labelled 'simplify'. A student would learn a wrong picture of rationalizing.

qwen3.6:27b-mlx: uphold — Line 2 multiplies top and bottom by sqrt(x+1), not the conjugate sqrt(x+1)+1 that its note describes. It is equal to line 1, but it does not rationalize anything, so the 'rationalize' label names a rule the step did not apply, and the jump from line 2 to line 3 then hides the real conjugate multiplication inside a step labelled 'simplify'.

Every verdict on record (7)

claude-sonnet-5-5: fail (misleading) 2026-10-04 — The final answer 1/2 is correct and every line is equal to the problem. However, line 2 does not show the conjugate multiplication it claims, and line 3 then does the conjugate multiplication and the numerator simplification in one step labelled 'simplify'. A student would learn a wrong picture of rationalizing.
qwen3.6:27b-mlx: fail (error) 2026-10-03 — Step 2 claims to multiply by the conjugate, but the expression shown multiplies by sqrt(x+1) instead of (sqrt(x+1)+1). This is a mathematical error in the step's content, not just a labeling issue.
gpt-oss:20b: pass 2026-10-03
qwen3.6:27b-mlx: fail (error) 2026-09-29 — Step 2 is mathematically incorrect: it multiplies by sqrt(x+1) instead of the conjugate (sqrt(x+1)+1), resulting in an expression not equal to the previous line.
gpt-oss:20b: fail (error) 2026-09-29 — Step 2 incorrectly applies the rationalization: it multiplies by √(x+1) instead of the conjugate √(x+1)+1, altering the expression in two ways (changing the numerator and the denominator) in a single step. This violates the rule that each step must change only one thing and use a correct rule.
qwen3.6:27b-mlx: fail (error) 2026-09-29 — Step 2 claims to multiply by the conjugate (sqrt(x+1) + 1) but actually multiplies by sqrt(x+1), which is algebraically incorrect for rationalizing the numerator. The resulting expression in Step 2 does not match the stated operation, and the transition to Step 3 is invalid based on the text of Step 2.
gpt-oss:20b: fail (error) 2026-09-29 — Step 2 incorrectly applies the rationalization: it multiplies by √(x+1) instead of the conjugate √(x+1)+1, so the algebraic transformation is invalid.

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.