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

Problem 1.136 · easy

1. \[ \lim_{x \to -1^+}\left(\frac{\tan{\left(x + 1 \right)}}{x + 1}\right) \]
   limitIdentify the limit to be evaluated.✓ Proved
2. \[ = \lim_{x \to -1^+}\left(\frac{\sin{\left(x + 1 \right)}}{\left(x + 1\right) \cos{\left(x + 1 \right)}}\right) \]
   rewriteRewrite the tangent function in terms of sine and cosine.✓ Proved
3. \[ = \lim_{x \to -1^+}\left(\frac{\sin{\left(x + 1 \right)}}{x + 1}\right) \lim_{x \to -1^+} \frac{1}{\cos{\left(x + 1 \right)}} \]
   limit-lawApply the product rule for limits.✓ Proved
4. \[ = \lim_{x \to -1^+} \frac{1}{\cos{\left(x + 1 \right)}} \]
   limitEvaluate the first limit using the fundamental trigonometric limit.✓ Proved
5. \[ = 1 \]
   limit simplify simplifyEvaluate the limit by direct substitution. Simplify the cosine value. Final result.✓ Proved