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

Problem 1.326 · medium

1. \[ \lim_{x \to 1^+}\left(\frac{8 x^{3} - 8}{2 x - 2}\right) \]
   limit factorStart with the limit of the function as x approaches 1 from the right. Factor out the constant 8 from the numerator.✓ Proved
2. \[ = \lim_{x \to 1^+}\left(\frac{\left(8 x - 8\right) \left(x^{2} + x + 1\right)}{2 x - 2}\right) \]
   factor algebraFactor the difference of cubes (x^3 - 1). Factor the denominator.✓ Proved
3. \[ = \lim_{x \to 1^+}\left(4 x^{2} + 4 x + 4\right) \]
   cancel simplifyCancel the common factor (x - 1). Simplify the remaining fraction.✓ Proved
4. \[ = \lim_{x \to 1^+} 12 \]
   substitutionEvaluate the expression at x = 1.✓ Proved
5. \[ = 12 \]
   limitCalculate the final value.✓ Proved