Limit of \( \displaystyle \frac{\sin{\left(5 x + 5 \right)}}{\sin{\left(2 x + 2 \right)}} \) as \( x \to -1 \)

Problem 1.413 · medium

1. \[ \lim_{x \to -1^+}\left(\frac{\sin{\left(5 x + 5 \right)}}{\sin{\left(2 x + 2 \right)}}\right) \]
   limit factorEvaluate the limit of the given function. Factor the arguments of the sine functions.✓ Proved
2. \[ = \lim_{x \to -1^+}\left(\frac{\frac{d}{d x} \sin{\left(5 x + 5 \right)}}{\frac{d}{d x} \sin{\left(2 x + 2 \right)}}\right) \]
   lhopitalThe limit is an indeterminate form 0/0, so apply L'Hopital's rule.✓ Proved
3. \[ = \lim_{x \to -1^+}\left(\frac{5 \cos{\left(5 x + 5 \right)}}{2 \cos{\left(2 x + 2 \right)}}\right) \]
   simplifyCompute the derivatives of the numerator and denominator.✓ Proved
4. \[ = \lim_{x \to -1^+} \frac{5}{2} \]
   substitution simplify simplifySubstitute the value x = -1 into the expression. Evaluate the cosine terms. Simplify the numerical fraction.✓ Proved
5. \[ = \frac{5}{2} \]
   limitThe limit is evaluated.✓ Proved