# 微積分學/極限/習題

 ← 極限/一些極限性質的證明 微積分學 導數 → 極限/習題

## 基礎題

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：${\displaystyle 25}$

## 單側極限

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

## 雙側極限

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

## 無窮極限

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：${\displaystyle 0}$

## 分段函數極限

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：${\displaystyle 1}$
 ← 極限/一些極限性質的證明 微積分學 導數 → 極限/習題