Exercise · ZW-SPECIAL-TRIG-LIMITS-010 · ★★☆ Intermediate
Compute \(\lim_{x\to 0}\,\frac{\tan {2x}}{\sin {5x}}\).
Choose one option.
Show answer
- D. \(\frac{2}{5}\)
Explanation
Since \(\tan 2x=\dfrac{\sin 2x}{\cos 2x}\), the quotient is \(\dfrac{\sin 2x}{\sin 5x}\cdot\dfrac{1}{\cos 2x}\to\dfrac{2}{5}\cdot1\cdot1=\dfrac{2}{5}\).
Review the concept: Special trigonometric limits
Two basic limits in radians: \(\frac{\sin x}{x}\) tends to \(1\) and \(\frac{1-\cos x}{x}\) tends to \(0\) as \(x\) tends to \(0\).
\[\begin{gathered}\lim_{x\to0}\frac{\sin x}{x}=1\\ \lim_{x\to0}\frac{1-\cos x}{x}=0\end{gathered}\]
More practice problems: special trigonometric limits
- ZW-SPECIAL-TRIG-LIMITS-001 Compute lim _x→ 0 (sin 3 x)/(x).
- ZW-SPECIAL-TRIG-LIMITS-002 Compute lim _x→ 0 (sin 5 x)/(2 x).
- ZW-SPECIAL-TRIG-LIMITS-003 Compute lim _x→ 0 (tan 4 x)/(x).
- ZW-SPECIAL-TRIG-LIMITS-004 Compute lim _x→ 0 (1 - cos x)/(x^2).
- ZW-SPECIAL-TRIG-LIMITS-005 Compute lim _x→ 0 (sin 2 x)/(sin 3 x).
- ZW-SPECIAL-TRIG-LIMITS-006 Compute lim _x→ 0 (sin 7x)/(x).
- ZW-SPECIAL-TRIG-LIMITS-007 Compute lim _x→ 0 (sin 4x)/(8x).
- ZW-SPECIAL-TRIG-LIMITS-008 Compute lim _x→ 0 (1-cos 2x)/(x^2).