Đề bài

Biết \(\mathop {\lim }\limits_{x \to 0} \frac{{\sin x}}{x} = 1\). Hãy tính: a) \(\mathop {\lim }\limits_{x \to 0} \frac{{\sin x}}{{{x^3}}}\);                                   b) \(\mathop {\lim }\limits_{x \to {0^ + }} \frac{{\sin x}}{{{x^2}}}\)                               c) \(\mathop {\lim }\limits_{x \to {0^ - }} \frac{{\sin x}}{{{x^2}}}\).

Lời giải chi tiết

Đặt \(f(x) = \frac{{\sin x}}{x}\). Khi đóa) \(\mathop {\lim }\limits_{x \to 0} \frac{{\sin x}}{{{x^3}}} = \mathop {\lim }\limits_{x \to 0} \frac{{f(x)}}{{{x^2}}} =  + \infty .\)b) \(\mathop {\lim }\limits_{x \to {0^ + }} \frac{{\sin x}}{{{x^2}}} = \mathop {\lim }\limits_{x \to {0^ + }} \frac{{f(x)}}{x} =  + \infty \).c) \(\mathop {\lim }\limits_{x \to {0^ - }} \frac{{\sin x}}{{{x^2}}} = \mathop {\lim }\limits_{x \to {0^ - }} \frac{{f(x)}}{x} =  - \infty .\)