Limit of \( \displaystyle \frac{8 x^{3} - 1}{4 x^{2} - 1} \) as \( x \to \frac{1}{2} \)

Problem 1.133 · medium

1. \[ \lim_{x \to \frac{1}{2}^+}\left(\frac{8 x^{3} - 1}{4 x^{2} - 1}\right) \]
   limitIdentify the limit to be evaluated.✓ Proved
2. \[ = \lim_{x \to \frac{1}{2}^+}\left(\frac{4 x^{2} + 2 x + 1}{2 x + 1}\right) \]
   factor cancelFactor the numerator and denominator. Cancel the common factor (2*x - 1).✓ Proved
3. \[ = \lim_{x \to \frac{1}{2}^+}\left(4 x^{2} + 2 x + 1\right) \left(\lim_{x \to \frac{1}{2}^+}\left(2 x + 1\right)\right)^{-1} \]
   limit-lawApply the quotient rule for limits.✓ Proved
4. \[ = \frac{3}{2} \]
   limit simplifyEvaluate the limits of the numerator and denominator. Simplify the resulting numerical expression.✓ Proved