# 练习题（一）

求极限：

1. $\Large \lim\limits_{x \to 2}{\frac{x^2+5}{x-3}}=-9$

2. $\Large \lim\limits_{x \to \sqrt{3}}{\frac{x^2-3}{x^2+1}}=0$

3. $\Large \lim\limits_{x \to 1}{\frac{x^2-2x+1}{x^2-1}}=0$

4. $\Large \lim\limits_{x \to 0}{\frac{4x^3-2x^2+x}{3x^2+2x}}=\frac{1}{2}$

5. $\Large \lim\limits_{h \to 0}{\frac{(x+h)^2-x^2}{h}}=2x$

6. $\Large \lim\limits_{x \to \infty}{(2-\frac{1}{x}+\frac{1}{x^2})}=2$

7. $\Large \lim\limits_{x \to \infty}{\frac{x^2-1}{2x^2-x-1}}$

8. $\Large \lim\limits_{x \to \infty}{\frac{x^2+x}{x^4-3x^2+1}}$

9. $\Large \lim\limits_{x \to 4}{\frac{x^2-6x+8}{x^2-5x+4}}$

10. $\Large \lim\limits_{x \to \infty}{(1+\frac{1}{x})(2-\frac{1}{x^2})}$

11. $\Large \lim\limits_{x \to \infty}{(1+\frac{1}{2}+\frac{1}{4}+...+\frac{1}{2^n})}$

12. $\Large \lim\limits_{x \to \infty}{\frac{1+2+3...+(n-1)}{n^2}}$

13. $\Large \lim\limits_{x \to \infty}{\frac{(n+1)(n+2)(n+3)}{5n^3}}$

14. $\Large \lim\limits_{x \to 1}{(\frac{1}{1-x}-\frac{3}{1-x^3})}$

15. $\Large \lim\limits_{x \to 2}{\frac{x^3+2x^2}{(x-2)^2}}$

16. $\Large \lim\limits_{x \to \infty}{\frac{x^2}{2x+1}}$

17. $\Large \lim\limits_{x \to \infty}{(2x^3-x+1)}$

18. $\Large \lim\limits_{x \to 0}{x^2\sin{\frac{1}{x}}}$

19. $\Large \lim\limits_{x \to \infty}{\frac{\arctan x}{x}}$

20. $\Large \lim\limits_{x \to \infty}{\frac{2x+1}{x}}$

21. $\Large \lim\limits_{x \to 0}{\frac{1-x^2}{1-x}}$

22. $\Large \lim\limits_{x \to 0}{\frac{\sin{\omega x}}{x}}$

23. $\Large \lim\limits_{x \to 0}{\frac{\tan 3x}{x}}$

24. $\Large \lim\limits_{x \to 0}{\frac{\sin 2x}{\sin 5x}}$

25. $\Large \lim\limits_{x \to 0}{x\cot x}$

26. $\Large \lim\limits_{x \to 0}{\frac{1-\cos 2x}{x\sin x}}$

27. $\Large \lim\limits_{x \to \infty}{2^n\sin \frac{x}{2^n}}$，x 为不等于0的常数

28. $\Large \lim\limits_{x \to 0}{(1-x)^{\frac{1}{x}}}$

29. $\Large \lim\limits_{x \to 0}{(1+2x)^{\frac{1}{x}}}$

30. $\Large \lim\limits_{x \to \infty}{(\frac{1+x}{x})^{2x}}$

31. $\Large \lim\limits_{x \to \infty}{(1-\frac{1}{x})^{kx}}$，k 为正整数

32. $\Large \lim\limits_{x \to 0}{\frac{\tan 3x}{2x}}$

33. $\Large \lim\limits_{x \to 0}{\frac{\sin x^n}{\sin^m x}}$，m、n 为正整数

34. $\Large \lim\limits_{x \to 0}{\frac{\tan x -\sin x}{\sin^3x}}$

35. $\Large \lim\limits_{x \to 0}{\frac{\sin x-\tan x}{(\sqrt[3]{1+x^2}-1)(\sqrt{1+\sin x}-1)}}$

36. $\Large \lim\limits_{x \to 0}{\sqrt{x^2-2x+5}}$

37. $\Large \lim\limits_{\alpha \to \frac{\pi}{4}}{\sin^3 2\alpha}$

38. $\Large \lim\limits_{x \to \frac{\pi}{6}}{\ln(2\cos 2x)}$

39. $\Large \lim\limits_{x \to 0}{\frac{\sqrt{x+1}-1}{x}}$

40. $\Large \lim\limits_{x \to 1}{\frac{\sqrt{5x-4}-\sqrt{x}}{x-1}}$

41. $\Large \lim\limits_{x \to \alpha}{\frac{\sin x-\sin \alpha}{x-\alpha}}$

42. $\Large \lim\limits_{x \to +\infty}{(\sqrt{x^2+x}-\sqrt{x^2-x})}$

43. $\Large \lim\limits_{x \to \infty}{e^{\frac{1}{x}}}$

44. $\Large \lim\limits_{x \to 0}{\ln{\frac{\sin x}{x}}}$

45. $\Large \lim\limits_{x \to \infty}{(1+\frac{1}{x})^{\frac{x}{2}}}$

46. $\Large \lim\limits_{x \to 0}{(1+3\tan^2x)^{\cot^2x}}$

47. $\Large \lim\limits_{x \to \infty}{(\frac{3+x}{6+x})^{\frac{x-1}{2}}}$

48. $\Large \lim\limits_{x \to 0}{\frac{\sqrt{1+\tan x}-\sqrt{1+\sin x}}{x\sqrt{1+\sin^2x}-x}}$

49. $\Large \lim\limits_{x \to 1}{\frac{x^2-x+1}{(x-1)^2}}$

50. $\Large \lim\limits_{x \to +\infty}{x(\sqrt{x^2+1}-x)}$

51. $\Large \lim\limits_{x \to \infty}{(\frac{2x+3}{2x+1})^{x+1}}$

52. $\Large \lim\limits_{x \to 0}{\frac{\tan x-\sin x}{x^3}}$

53. $\Large \lim\limits_{x \to 0}{(\frac{a^x+b^x+c^x}{3})^{\frac{1}{x}}}$，a、b、c 都大于 0

54. $\Large \lim\limits_{x \to \frac{\pi}{2}}{(\sin x)^{\tan x}}$