Limit of \( \displaystyle \frac{\sin{\left(9 x \right)}}{3 x} \) as \( x \to 0 \)

Problem 1.225 · easy

1. \[ \lim_{x \to 0^+}\left(\frac{\sin{\left(9 x \right)}}{3 x}\right) \]
   limit algebra algebra algebra algebra limit-lawStart with the original limit expression. Separate the constant factor 1/3 from the fraction. Rearrange the expression for clarity. Multiply and divide by 9 to create the standard limit form sin(u)/u. Simplify the constant coefficients. Apply the limit law for a constant times a function.✓ Proved
2. \[ = 3 \lim_{x \to 0^+}\left(\frac{\sin{\left(9 x \right)}}{9 x}\right) \]
   limit-lawPull the constant 3 out of the limit.✓ Proved
3. \[ = 3 \]
   trig-limit simplifyUse the fundamental trigonometric limit lim_{u->0} sin(u)/u = 1, where u = 9x. Evaluate the final multiplication.✓ Proved