Exercise · ZW-SPECIAL-TRIG-LIMITS-011 · ★★★ Advanced
Compute \(\lim_{x\to 0}\,\frac{\sin {x^{2}}}{1-\cos {x}}\).
Choose one option.
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- C. \(2\)
Explanation
Write \(\dfrac{\sin x^{2}}{1-\cos x}=\dfrac{\sin x^{2}}{x^{2}}\cdot\dfrac{x^{2}}{1-\cos x}\). The first factor tends to 1 and, since \(\dfrac{1-\cos x}{x^{2}}\to\dfrac{1}{2}\), the second tends to 2. The limit is 2.
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).