Cho hàm số f(x) = sin x. Tính lim h tới 0 f ( x +h) - f(x) /h

8/22

Cho hàm số \(f\left( x \right) = \sin x\). Tính \(\mathop {\lim }\limits_{h \to 0} \frac{{f\left( {x + h} \right) - f\left( x \right)}}{h}\)

\(\cos \frac{x}{2}\).

\(2\sin \frac{x}{2}\).

\(\cos x\).

\( - \cos x\).

Giải thích

Ta có: \(\mathop {\lim }\limits_{h \to 0} \frac{{\sin \left( {x + h} \right) - \sin x}}{h} = \mathop {\lim }\limits_{h \to 0} \frac{{2\cos \left( {x + \frac{h}{2}} \right)\sin \frac{h}{2}}}{h} = \mathop {\lim }\limits_{h \to 0} \frac{{\sin \frac{h}{2}}}{{\frac{h}{2}}}\cos \left( {x + \frac{h}{2}} \right) = 1.\cos x = \cos x\)

Vì:\(\mathop {\lim }\limits_{h \to 0} \frac{{\sin \frac{h}{2}}}{{\frac{h}{2}}} = 1\).