Limit of \( \displaystyle \frac{\tan{\left(x \right)}}{x} \) as \( x \to 0 \)

Problem 1.64 · easy

1. \[ \lim_{x \to 0^+}\left(\frac{\tan{\left(x \right)}}{x}\right) \]
   limitIdentify the limit to be evaluated.✓ Proved
2. \[ = \lim_{x \to 0^+}\left(\frac{\sin{\left(x \right)}}{x \cos{\left(x \right)}}\right) \]
   rewrite algebraRewrite tan(x) as sin(x)/cos(x). Rearrange the fraction.✓ Proved
3. \[ = \lim_{x \to 0^+}\left(\frac{\sin{\left(x \right)}}{x}\right) \lim_{x \to 0^+} \frac{1}{\cos{\left(x \right)}} \]
   limit-lawApply the product rule for limits.✓ Proved
4. \[ = \lim_{x \to 0^+} \frac{1}{\cos{\left(x \right)}} \]
   limitEvaluate the first limit using the fundamental trigonometric limit.✓ Proved
5. \[ = 1 \]
   limit simplify simplifyEvaluate the second limit by direct substitution. Simplify the numerical expression. Final result.✓ Proved