GIỚI HẠN $\underset{x\to 0}{\mathop{\lim }}\,\frac{\sin x}{x}=1$

A/ Công thức cần nhớ:

$$1-\cos \alpha =2{{\sin }^{2}}\frac{\alpha }{2}$$ ; $$\cos \alpha -\cos \beta =-2\sin \frac{\alpha +\beta }{2}\sin \frac{\alpha -\beta }{2}$$

B/ Bài tập mẫu cơ bản:

1)

$$\underset{x\to 0}{\mathop{\lim }}\,\frac{\sin 3x}{x}=\underset{x\to 0}{\mathop{\lim }}\,3.\frac{\sin 3x}{3x}=3.\underset{x\to 0}{\mathop{\lim }}\,\frac{\sin 3x}{3x}=3.1=3$$

$$\underset{x\to 0}{\mathop{\lim }}\,\frac{\sin ax}{x}=a;\left( a\ne 0 \right)$$

2)

$$\underset{x\to 0}{\mathop{\lim }}\,\frac{\sin 3x}{\sin 5x}=\underset{x\to 0}{\mathop{\lim }}\,\frac{\frac{\sin 3x}{x}}{\frac{\sin 5x}{x}}=\underset{x\to 0}{\mathop{\lim }}\,\frac{3}{5}.\frac{\frac{\sin 3x}{3x}}{\frac{\sin 5x}{5x}}=\frac{3}{5}.\underset{x\to 0}{\mathop{\lim }}\,\frac{\frac{\sin 3x}{3x}}{\frac{\sin 5x}{5x}}=\frac{3}{5}$$

$$\underset{x\to 0}{\mathop{\lim }}\,\frac{\sin ax}{\sin bx}=\frac{a}{b}$$

3)

$$\underset{x\to 0}{\mathop{\lim }}\,\frac{1-\cos x}{{{x}^{2}}}=\underset{x\to 0}{\mathop{\lim }}\,\frac{2{{\sin }^{2}}\frac{x}{2}}{{{x}^{2}}}=2.\underset{x\to 0}{\mathop{\lim }}\,\frac{\sin \frac{x}{2}}{x}.\frac{\sin \frac{x}{2}}{x}=2.\underset{x\to 0}{\mathop{\lim }}\,\frac{\sin \frac{x}{2}}{2.\frac{x}{2}}.\frac{\sin \frac{x}{2}}{2.\frac{x}{2}}=2.\frac{1}{4}=\frac{1}{2}$$

4)

$$\underset{x\to 0}{\mathop{\lim }}\,\frac{tgax}{bx}=\underset{x\to 0}{\mathop{\lim }}\,\frac{\sin ax}{bx}.\frac{1}{\cos ax}=\underset{x\to 0}{\mathop{\lim }}\,\frac{a.\frac{\sin ax}{ax}}{b}.\frac{1}{\cos ax}=\frac{a}{b}.\underset{x\to 0}{\mathop{\lim }}\,\frac{\sin ax}{ax}.\frac{1}{\cos ax}=\frac{a}{b}.1.\frac{1}{\cos 0}=\frac{a}{b}$$

C/ Bài tập .

5)

$$\underset{x\to 0}{\mathop{\lim }}\,\frac{\cos 12x-\cos 10x}{\cos 8x-\cos 6x}$$

6)

$$\underset{x\to 0}{\mathop{\lim }}\,\frac{1-\cos x.\cos 2x}{{{x}^{2}}}$$

7)

$$\underset{x\to 0}{\mathop{\lim }}\,\frac{1-\sqrt{2x+1}}{\sin 2x}$$

8)

$$\underset{x\to 0}{\mathop{\lim }}\,\frac{1-\cos 5x}{tgx-\sin 2x}$$

9) ĐH Thương mại 99.

$$\underset{x\to 0}{\mathop{\lim }}\,\frac{\sqrt{1+{{x}^{2}}}-\cos x}{{{x}^{2}}}$$

10) ĐHQG 99, khối D.

$$\underset{x\to 1}{\mathop{\lim }}\,\frac{{{x}^{3}}+{{x}^{2}}-2}{\sin \left( x-1 \right)}$$

11) ĐH Bách khoa,D,2001.

$$\underset{x\to 0}{\mathop{\lim }}\,\frac{1-\sqrt{2{{x}^{2}}+1}}{1-\cos x}$$

12) ĐH Hàng hải 2001.

$$\underset{x\to 0}{\mathop{\lim }}\,\frac{{{\cos }^{4}}x-{{\sin }^{4}}x-1}{\sqrt{{{x}^{2}}+1}-1}$$

13) ĐH Hàng hải 2000.

$$\underset{x\to 0}{\mathop{\lim }}\,\frac{\sqrt{1+tgx}-\sqrt{1+\sin x}}{{{x}^{3}}}$$

14) ĐHSP 2,2000.

$$\underset{x\to \frac{\pi }{4}}{\mathop{\lim }}\,tg2x.tg\left( \frac{\pi }{4}-x \right)$$

15) ĐH AnNninh 2000.

$$\underset{x\to 0}{\mathop{\lim }}\,\frac{98}{83}\left( \frac{1-\cos 3x\cos 5x\cos 7x}{{{\sin }^{2}}7x} \right)$$

16)

$$\underset{x\to 0}{\mathop{\lim }}\,\frac{1-\cos 5x\cos 7x}{{{\sin }^{2}}11x}$$

17) Hàng Hải 97.

$$\underset{x\to {{0}^{+}}}{\mathop{\lim }}\,\frac{1-\sqrt{\cos x}}{1-\cos \sqrt{x}}$$

18) ĐHQG,Khối D,2000.

$$\underset{x\to 0}{\mathop{\lim }}\,\frac{\sqrt{2x+1}-\sqrt[3]{{{x}^{2}}+1}}{\sin x}$$

19)

$$\underset{x\to \frac{\pi }{4}}{\mathop{\lim }}\,\frac{\sqrt{2}-2\cos x}{\sin \left( x-\frac{\pi }{4} \right)}$$

20) CĐSP 2001.

$$\underset{x\to -n}{\mathop{\lim }}\,\frac{tg\pi x}{x+n};\left( n\in {{N}^{*}} \right)$$

21)

$$\underset{x\to \infty }{\mathop{\lim }}\,\left( x+2 \right)\sin \frac{3}{x}$$

22)ĐHQG 97.

$$\underset{x\to 0}{\mathop{\lim }}\,\frac{\left| 1-\left| 1+\sin 3x \right| \right|}{\sqrt{1-\cos x}}$$

23) TN 97.

$$\underset{x\to 0}{\mathop{\lim }}\,\frac{\cos \left( \frac{\pi }{2}\cos x \right)}{\sin \left( \frac{{{x}^{2}}}{2} \right)}$$

24)BCVT99.

$$\underset{x\to \infty }{\mathop{\lim }}\,\frac{x-\sin x}{x+\sin x}$$

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