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

Problem 1.54 · easy

1. \[ \lim_{x \to 1^+}\left(\frac{\tan{\left(x - 1 \right)}}{x - 1}\right) \]
   limitStart with the original limit.✓ Proved
2. \[ = \lim_{x \to 1^+}\left(\frac{\sin{\left(x - 1 \right)}}{\left(x - 1\right) \cos{\left(x - 1 \right)}}\right) \]
   rewrite algebraUse the identity tan(u) = sin(u)/cos(u). Rearrange the fraction.✓ 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. \[ = \lim_{x \to 1^+}\left(\frac{\frac{d}{d x} \sin{\left(x - 1 \right)}}{\frac{d}{d x} \left(x - 1\right)}\right) \lim_{x \to 1^+} \frac{1}{\cos{\left(x - 1 \right)}} \]
   lhopitalApply L'Hopital's rule to the first limit.✓ Proved
5. \[ = \lim_{x \to 1^+} \cos{\left(x - 1 \right)} \lim_{x \to 1^+} \frac{1}{\cos{\left(x - 1 \right)}} \]
   simplifyEvaluate the derivatives.✓ Proved
6. \[ = \lim_{x \to 1^+} 1 \]
   algebra simplifyCombine the terms. Simplify the expression inside the limit.✓ Proved
7. \[ = 1 \]
   limitEvaluate the final limit.✓ Proved