Limit of \( \displaystyle \frac{\sin{\left(4 x + 4 \right)}}{3 x + 3} \) as \( x \to -1 \)

Problem 1.165 · medium

1. \[ \lim_{x \to -1^+}\left(\frac{\sin{\left(4 x + 4 \right)}}{3 x + 3}\right) \]
   limit factorIdentify the limit to evaluate. Factor the argument of the sine function.✓ Proved
2. \[ = \lim_{x \to -1^+}\left(\frac{\frac{d}{d x} \sin{\left(4 x + 4 \right)}}{\frac{d}{d x} \left(3 x + 3\right)}\right) \]
   lhopitalThe limit is an indeterminate form 0/0, so apply L'Hopital's rule.✓ Proved
3. \[ = \lim_{x \to -1^+}\left(\frac{4 \cos{\left(4 x + 4 \right)}}{3}\right) \]
   simplifyCompute the derivatives of the numerator and denominator.✓ Proved
4. \[ = \frac{4}{3} \]
   limit simplifyEvaluate the limit by substituting x = -1. Simplify the final numerical value.✓ Proved