"limx→0sin⁡h−sin⁡xx(cos⁡h(x)−cos⁡(x))" was answered by our experts

High School
Geometry
Trigonometry
Trigonometric equations and identities

Roger Smith

Answered question

2021-12-30

lim_{x \to 0} \frac{\sin h - \sin x}{x (\cos h (x) - \cos (x))}

Answer & Explanation

Buck Henry

Beginner2021-12-31Added 33 answers

Hint: In fact,

L = lim_{x \to 0} \frac{\sin h (x) - \sin (x)}{x (\cos h (x) - \cos (x))}

= lim_{x \to 0} \frac{\sin h (x) - \sin (x)}{x^{3}} \cdot lim_{x \to 0} \frac{x^{2}}{\cos h (x) - \cos (x)}

= lim_{x \to 0} \frac{\cos h (x) - \cos (x)}{3 x^{2}} \cdot lim_{x \to 0} \frac{2 x}{\sin h (x) + \sin (x)}

Dabanka4v

Beginner2022-01-01Added 36 answers

Since

\sin h x - \sin x \sim \frac{1}{3} x^{3}

while

\cos h x - \cos x \sim x^{2}

, the limit is

\frac{1}{3}

Vasquez

Expert2022-01-09Added 669 answers

$\begin{matrix} lim_{x \to 0} \frac{\sin h (x) - \sin (x)}{x (\cos h (x) - \cos (x))} \\ = lim_{x \to 0} \frac{(x + \frac{x^{6}}{6} + o (x^{4})) - (x - \frac{x^{3}}{6} + o (x^{4}))}{x ((1 + \frac{x^{2}}{2} + o (x^{3}))) - (1 - \frac{x^{2}}{2} + o (x^{3}))} \\ = lim_{x \to 0} \frac{\frac{x^{3}}{3} + o (x^{4})}{x + o (x^{4})} \\ = lim_{x \to 0} \frac{\frac{1}{3} + o (x^{3})}{1 + o (x^{3})} \\ = \frac{1}{3} \end{matrix}$

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