Home›Calculus 1›Limits by rationalizing›Problem 1.798

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

Problem 1.798 · medium Video

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

\[ \lim_{x \to 2^+}\left(\frac{2 x - 4}{\sqrt{2} \sqrt{x} - 2}\right) \]
limit factorStart with the original limit. Factor out 2 from the numerator.✓ Proved
\[ = \lim_{x \to 2^+}\left(\frac{\sqrt{2} \left(2 x - 4\right)}{2 \left(\sqrt{x} - \sqrt{2}\right)}\right) \]
algebraFactor out sqrt(2) from the denominator.✓ Proved
\[ = \lim_{x \to 2^+}\left(\frac{\sqrt{2} \left(\sqrt{x} + \sqrt{2}\right) \left(2 x - 4\right)}{2 \left(x - 2\right)}\right) \]
rationalizeMultiply numerator and denominator by the conjugate.✓ Proved
\[ = \lim_{x \to 2^+}\left(\frac{\sqrt{2} \left(2 \sqrt{x} + 2 \sqrt{2}\right)}{2}\right) \]
cancelCancel the common factor (x - 2).✓ Proved
\[ = \lim_{x \to 2^+}\left(\sqrt{2} \left(\sqrt{x} + \sqrt{2}\right)\right) \]
simplify simplifySimplify the constant coefficient. Simplify 2/sqrt(2) to sqrt(2).✓ Proved
\[ = \lim_{x \to 2^+}\left(\sqrt{2} \sqrt{x} + 2\right) \]
simplifyDistribute sqrt(2) into the parentheses.✓ Proved
\[ = 4 \]
limit simplify simplifyEvaluate the limit by substituting x = 2. Simplify sqrt(2)*sqrt(2). Final result.✓ Proved

Answer \( 4 \)

The graph of f(x) = (2x - 4)/(sqrt(2)·sqrt(x) - 2). Two points slide along the curve toward x = 2, one from each side; their heights close in on 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 sqrt(2)*sqrt(x) - 2 = 0
3	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where sqrt(2)*sqrt(x) - 2 = 0 undefined where sqrt(x) - sqrt(2) = 0
4	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where sqrt(x) - sqrt(2) = 0 undefined where x - 2 = 0
5	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where x - 2 = 0
6	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 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
9	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0
10	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0
11	✓ 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 factoring, rationalization, and cancellation to resolve the indeterminate form. Each step isolates a single algebraic transformation, and the labels accurately describe the operations performed.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-11-03 — The solution correctly applies factoring, rationalization, and cancellation to resolve the indeterminate form. Each step isolates a single algebraic transformation, and the labels accurately describe the operations performed.
gpt-oss:20b: pass 2026-11-03
qwen3.6:27b-mlx: pass 2026-11-02
gpt-oss:20b: pass 2026-11-02
qwen3.6:27b-mlx: pass 2026-11-02 — The solution correctly applies algebraic manipulations 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 2026-11-02

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 2026-11-03 with SymPy 1.14.0.