Limit of \( \displaystyle \frac{\sqrt{2} \sqrt{x} - 1}{2 x - 1} \) as \( x \to \frac{1}{2} \)

Problem 1.224 · medium

1. \[ \lim_{x \to \frac{1}{2}^+}\left(\frac{\sqrt{2} \sqrt{x} - 1}{2 x - 1}\right) \]
   limitStart with the limit of the original function.✓ Proved
2. \[ = \lim_{x \to \frac{1}{2}^+}\left(\sqrt{2} \sqrt{x} - 1\right) \left(\lim_{x \to \frac{1}{2}^+}\left(2 x - 1\right)\right)^{-1} \]
   limit-lawCheck the limit of the numerator and denominator separately.Not checked
3. \[ = 1 \]
   limitEvaluate the limits of the numerator and denominator.Not checked
4. \[ = \lim_{x \to \frac{1}{2}^+} \text{NaN} \]
   simplifyThe limit results in an indeterminate form.Not checked
5. \[ = \lim_{x \to \frac{1}{2}^+}\left(\frac{\frac{d}{d x} \left(\sqrt{2} \sqrt{x} - 1\right)}{\frac{d}{d x} \left(2 x - 1\right)}\right) \]
   lhopitalApply L'Hopital's rule.Not checked
6. \[ = \lim_{x \to \frac{1}{2}^+}\left(\frac{\sqrt{2}}{4 \sqrt{x}}\right) \]
   simplify algebraCompute the derivatives. Simplify the fraction.✓ Proved
7. \[ = \frac{1}{2} \]
   limit simplify algebra simplify simplifyEvaluate the limit by substituting x = 1/2. Simplify the denominator. Multiply the numerator and denominator by sqrt(2). Simplify the expression. Final answer.✓ Proved