# 微积分学/极限/习题

 ← 极限/一些极限性质的证明 微积分学 导数 → 极限/习题

## 基础题

1. ${\displaystyle \lim _{x\to 2}{\Big (}4x^{2}-3x+1{\Big )}}$ …………答案：${\displaystyle 11}$
2. ${\displaystyle \lim _{x\to 5}x^{2}}$ …………答案：${\displaystyle 25}$

## 单侧极限

1. ${\displaystyle \lim _{x\to 0^{-}}{\frac {x^{3}+x^{2}}{x^{3}+2x^{2}}}}$ …………答案：${\displaystyle {\frac {1}{2}}}$
2. ${\displaystyle \lim _{x\to 7^{-}}{\Big (}|x^{2}+x|-x{\Big )}}$ …………答案：${\displaystyle 49}$
3. ${\displaystyle \lim _{x\to -1^{-}}{\sqrt {1-x^{2}}}}$ …………答案：${\displaystyle 0}$
4. ${\displaystyle \lim _{x\to -1^{+}}{\sqrt {1-x^{2}}}}$ …………答案：不存在

## 双侧极限

1. ${\displaystyle \lim _{x\to -1}{\frac {1}{x-1}}}$ …………答案：${\displaystyle -{\frac {1}{2}}}$
2. ${\displaystyle \lim _{x\to 4}{\frac {1}{x-4}}}$ …………答案：不存在
3. ${\displaystyle \lim _{x\to 2}{\frac {1}{x-2}}}$ …………答案：不存在
4. ${\displaystyle \lim _{x\to -3}{\frac {x^{2}-9}{x+3}}}$ …………答案：${\displaystyle -6}$
5. ${\displaystyle \lim _{x\to 3}{\frac {x^{2}-9}{x-3}}}$ …………答案：${\displaystyle 6}$
6. ${\displaystyle \lim _{x\to -1}{\frac {x^{2}+2x+1}{x+1}}}$ …………答案：${\displaystyle 0}$
7. ${\displaystyle \lim _{x\to -1}{\frac {x^{3}+1}{x+1}}}$ …………答案：${\displaystyle 3}$
8. ${\displaystyle \lim _{x\to 4}{\frac {x^{2}+5x-36}{x^{2}-16}}}$ …………答案：${\displaystyle {\frac {13}{8}}}$
9. ${\displaystyle \lim _{x\to 25}{\frac {x-25}{{\sqrt {x}}-5}}}$ …………答案：${\displaystyle 10}$
10. ${\displaystyle \lim _{x\to 0}{\frac {|x|}{x}}}$ …………答案：不存在
11. ${\displaystyle \lim _{x\to 2}{\frac {1}{(x-2)^{2}}}}$ …………答案：${\displaystyle +\infty }$
12. ${\displaystyle \lim _{x\to 3}{\frac {\sqrt {x^{2}+16}}{x-3}}}$ …………答案：不存在
13. ${\displaystyle \lim _{x\to -2}{\frac {3x^{2}-8x-3}{2x^{2}-18}}}$ …………答案：${\displaystyle -{\frac {5}{2}}}$
14. ${\displaystyle \lim _{x\to 2}{\frac {x^{2}+2x+1}{x^{2}-2x+1}}}$ …………答案：${\displaystyle 9}$
15. ${\displaystyle \lim _{x\to 3}{\frac {x+3}{x^{2}-9}}}$ …………答案：不存在
16. ${\displaystyle \lim _{x\to -1}{\frac {x+1}{x^{2}+x}}}$ …………答案：${\displaystyle -1}$
17. ${\displaystyle \lim _{x\to 1}{\frac {1}{x^{2}+1}}}$ …………答案：${\displaystyle {\frac {1}{2}}}$
18. ${\displaystyle \lim _{x\to 1}\left(x^{2}+5x-{\frac {1}{2-x}}\right)}$ …………答案：${\displaystyle 5}$
19. ${\displaystyle \lim _{x\to 1}{\frac {x^{2}-1}{x^{2}+2x-3}}}$ …………答案：${\displaystyle {\frac {1}{2}}}$
20. ${\displaystyle \lim _{x\to 1}{\frac {5x}{x^{2}+2x-3}}}$ …………答案：不存在

## 无穷极限

1. ${\displaystyle \lim _{x\to \infty }{\frac {-x+\pi }{x^{2}+3x+2}}}$ …………答案：${\displaystyle 0}$
2. ${\displaystyle \lim _{x\to -\infty }{\frac {x^{2}+2x+1}{3x^{2}+1}}}$ …………答案：${\displaystyle {\frac {1}{3}}}$
3. ${\displaystyle \lim _{x\to -\infty }{\frac {3x^{2}+x}{2x^{2}-15}}}$ …………答案：${\displaystyle {\frac {3}{2}}}$
4. ${\displaystyle \lim _{x\to -\infty }{\Big (}3x^{2}-2x+1{\Big )}}$ …………答案：${\displaystyle +\infty }$
5. ${\displaystyle \lim _{x\to \infty }{\frac {2x^{2}-32}{x^{3}-64}}}$ …………答案：${\displaystyle 0}$
6. ${\displaystyle \lim _{x\to \infty }6}$ …………答案：${\displaystyle 6}$
7. ${\displaystyle \lim _{x\to \infty }{\frac {3x^{2}+4x}{x^{4}+2}}}$ …………答案：${\displaystyle 0}$
8. ${\displaystyle \lim _{x\to -\infty }{\frac {2x+3x^{2}+1}{2x^{2}+3}}}$ …………答案：${\displaystyle {\frac {3}{2}}}$
9. ${\displaystyle \lim _{x\to -\infty }{\frac {x^{3}-3x^{2}+1}{3x^{2}+x+5}}}$ …………答案：${\displaystyle -\infty }$
10. ${\displaystyle \lim _{x\to \infty }{\frac {x^{2}+2}{x^{3}-2}}}$ …………答案：${\displaystyle 0}$

## 分段函数极限

1. ${\displaystyle f(x)={\begin{cases}(x-2)^{2}&{\text{, }}x<2\\x-3&{\text{, }}x\geq 2\end{cases}}}$
1. ${\displaystyle \lim _{x\to 2^{-}}f(x)}$ …………答案：${\displaystyle 0}$
2. ${\displaystyle \lim _{x\to 2^{+}}f(x)}$ …………答案：${\displaystyle -1}$
3. ${\displaystyle \lim _{x\to 2}f(x)}$ …………答案：不存在
2. ${\displaystyle g(x)={\begin{cases}-2x+1&{\text{, }}x\leq 0\\x+1&{\text{, }}0
1. ${\displaystyle \lim _{x\to 4^{+}}g(x)}$ …………答案：${\displaystyle 18}$
2. ${\displaystyle \lim _{x\to 4^{-}}g(x)}$ …………答案：${\displaystyle 5}$
3. ${\displaystyle \lim _{x\to 0^{+}}g(x)}$ …………答案：${\displaystyle 1}$
4. ${\displaystyle \lim _{x\to 0^{-}}g(x)}$ …………答案：${\displaystyle 1}$
5. ${\displaystyle \lim _{x\to 0}g(x)}$ …………答案：${\displaystyle 1}$
6. ${\displaystyle \lim _{x\to 1}g(x)}$ …………答案：${\displaystyle 2}$
3. ${\displaystyle h(x)={\begin{cases}2x-3&{\text{, }}x<2\\8&{\text{, }}x=2\\-x+3&{\text{, }}x>2\end{cases}}}$
1. ${\displaystyle \lim _{x\to 0}h(x)}$ …………答案：${\displaystyle -3}$
2. ${\displaystyle \lim _{x\to 2^{-}}h(x)}$ …………答案：${\displaystyle 1}$
3. ${\displaystyle \lim _{x\to 2^{+}}h(x)}$ …………答案：${\displaystyle 1}$
4. ${\displaystyle \lim _{x\to 2}h(x)}$ …………答案：${\displaystyle 1}$
 ← 极限/一些极限性质的证明 微积分学 导数 → 极限/习题