# 微积分学/极限/解答

## 基础题

1. ${\displaystyle \lim _{x\to 2}(4x^{2}-3x+1)}$
解答：${\displaystyle 4(4)-2(3)+1=16-6+1=\mathbf {11} }$
2. ${\displaystyle \lim _{x\to 5}x^{2}}$
解答：${\displaystyle 5^{2}=\mathbf {25} }$

## 单侧极限

1. ${\displaystyle \lim _{x\to 0^{-}}{\frac {x^{3}+x^{2}}{x^{3}+2x^{2}}}}$
解答：分解因式：${\displaystyle {\frac {x^{2}}{x^{2}}}\cdot {\frac {x+1}{x+2}}}$，可知${\displaystyle x=0}$为一可去间断点，故极限为${\displaystyle \mathbf {\frac {1}{2}} }$
2. ${\displaystyle \lim _{x\to 7^{-}}(|x^{2}+x|-x)}$
解答：${\displaystyle |7^{2}+7|-7=\mathbf {49} }$
3. ${\displaystyle \lim _{x\to -1^{+}}{\sqrt {1-x^{2}}}}$
解答：${\displaystyle {\sqrt {1-x^{2}}}}$${\displaystyle x^{2}<1}$时有意义，故极限为${\displaystyle {\sqrt {1-1^{2}}}=\mathbf {0} }$
4. ${\displaystyle \lim _{x\to -1^{-}}{\sqrt {1-x^{2}}}}$
解答：${\displaystyle {\sqrt {1-x^{2}}}}$${\displaystyle x^{2}>1}$时无意义，故极限不存在

## 双侧极限

1. ${\displaystyle \lim _{x\to -1}{\frac {1}{x-1}}}$
解答：${\displaystyle \mathbf {-{\frac {1}{2}}} }$
2. ${\displaystyle \lim _{x\to 4}{\frac {1}{x-4}}}$
解答：${\displaystyle \lim _{x\to 4^{-}}{\frac {1}{x-4}}=-\infty }$
${\displaystyle \lim _{x\to 4^{+}}{\frac {1}{x-4}}=+\infty }$
极限不存在
3. ${\displaystyle \lim _{x\to 2}{\frac {1}{x-2}}}$
解答：${\displaystyle \lim _{x\to 2^{-}}{\frac {1}{x-2}}=-\infty }$
${\displaystyle \lim _{x\to 2^{+}}{\frac {1}{x-2}}=+\infty }$
极限不存在
4. ${\displaystyle \lim _{x\to -3}{\frac {x^{2}-9}{x+3}}}$
解答：${\displaystyle \lim _{x\to -3}{\frac {(x+3)(x-3)}{x+3}}=\lim _{x\to -3}x-3=-3-3=\mathbf {-6} }$
5. ${\displaystyle \lim _{x\to 3}{\frac {x^{2}-9}{x-3}}}$
解答：${\displaystyle \lim _{x\to 3}{\frac {(x-3)(x+3)}{x-3}}=\lim _{x\to 3}x+3=3+3=\mathbf {6} }$
6. ${\displaystyle \lim _{x\to -1}{\frac {x^{2}+2x+1}{x+1}}}$
解答：${\displaystyle \lim _{x\to -1}{\frac {(x+1)(x+1)}{x+1}}=\lim _{x\to -1}x+1=-1+1=\mathbf {0} }$
7. ${\displaystyle \lim _{x\to -1}{\frac {x^{3}+1}{x+1}}}$
解答：${\displaystyle \lim _{x\to -1}{\frac {(x^{2}-x+1)(x+1)}{x+1}}=\lim _{x\to -1}x^{2}-x+1=(-1)^{2}-(-1)+1=1+1+1=\mathbf {3} }$
8. ${\displaystyle \lim _{x\to 4}{\frac {x^{2}+5x-36}{x^{2}-16}}}$
解答：${\displaystyle \lim _{x\to 4}{\frac {(x-4)(x+9)}{(x-4)(x+4)}}=\lim _{x\to 4}{\frac {x+9}{x+4}}={\frac {4+9}{4+4}}=\mathbf {\frac {13}{8}} }$
9. ${\displaystyle \lim _{x\to 25}{\frac {x-25}{{\sqrt {x}}-5}}}$
解答：${\displaystyle \lim _{x\to 25}{\frac {({\sqrt {x}}-5)({\sqrt {x}}+5)}{{\sqrt {x}}-5}}=\lim _{x\to 25}({\sqrt {x}}+5)={\sqrt {25}}+5=5+5=\mathbf {10} }$
10. ${\displaystyle \lim _{x\to 0}{\frac {\left|x\right|}{x}}}$
解答：${\displaystyle \lim _{x\to 0^{-}}{\frac {\left|x\right|}{x}}=\lim _{x\to 0^{-}}{\frac {-x}{x}}=\lim _{x\to 0^{-}}-1=-1}$
${\displaystyle \lim _{x\to 0^{+}}{\frac {\left|x\right|}{x}}=\lim _{x\to 0^{+}}{\frac {x}{x}}=\lim _{x\to 0^{+}}1=1}$
极限不存在
11. ${\displaystyle \lim _{x\to 2}{\frac {1}{(x-2)^{2}}}}$
解答：当${\displaystyle x}$趋近于${\displaystyle 2}$时，分母趋近于${\displaystyle 0}$，故极限为${\displaystyle \mathbf {+\infty } }$
12. ${\displaystyle \lim _{x\to 3}{\frac {\sqrt {x^{2}+16}}{x-3}}}$
解答：当${\displaystyle x}$趋近于${\displaystyle 3}$时，分子趋近于${\displaystyle 5}$，分母趋近于${\displaystyle 0}$，但从左侧趋近时极限为${\displaystyle -\infty }$，从右侧趋近时极限为${\displaystyle +\infty }$，故极限不存在
13. ${\displaystyle \lim _{x\to -2}{\frac {3x^{2}-8x-3}{2x^{2}-18}}}$
解答：${\displaystyle {\frac {3(-2)^{2}-8(-2)-3}{2(-2)^{2}-18}}={\frac {3(4)+16-3}{2(4)-18}}={\frac {12+16-3}{8-18}}={\frac {25}{-10}}=\mathbf {-{\frac {5}{2}}} }$
14. ${\displaystyle \lim _{x\to 2}{\frac {x^{2}+2x+1}{x^{2}-2x+1}}}$
解答：${\displaystyle {\frac {2^{2}+2(2)+1}{2^{2}-2(2)+1}}={\frac {4+4+1}{4-4+1}}={\frac {9}{1}}=\mathbf {9} }$
15. ${\displaystyle \lim _{x\to 3}{\frac {x+3}{x^{2}-9}}}$
解答：${\displaystyle \lim _{x\to 3}{\frac {x+3}{(x+3)(x-3)}}=\lim _{x\to 3}{\frac {1}{x-3}}}$
${\displaystyle \lim _{x\to 3^{-}}{\frac {1}{x-3}}=-\infty }$
${\displaystyle \lim _{x\to 3^{+}}{\frac {1}{x-3}}=+\infty }$
极限不存在
16. ${\displaystyle \lim _{x\to -1}{\frac {x+1}{x^{2}+x}}}$
解答：${\displaystyle \lim _{x\to -1}{\frac {x+1}{x(x+1)}}=\lim _{x\to -1}{\frac {1}{x}}={\frac {1}{-1}}=\mathbf {-1} }$
17. ${\displaystyle \lim _{x\to 1}{\frac {1}{x^{2}+1}}}$
解答：${\displaystyle {\frac {1}{1^{2}+1}}={\frac {1}{1+1}}=\mathbf {\frac {1}{2}} }$
18. ${\displaystyle \lim _{x\to 1}x^{3}+5x-{\frac {1}{2-x}}}$
解答：${\displaystyle 1^{3}+5(1)-{\frac {1}{2-1}}=1+5-{\frac {1}{1}}=6-1=\mathbf {5} }$
19. ${\displaystyle \lim _{x\to 1}{\frac {x^{2}-1}{x^{2}+2x-3}}}$
解答：${\displaystyle \lim _{x\to 1}{\frac {(x-1)(x+1)}{(x-1)(x+3)}}=\lim _{x\to 1}{\frac {x+1}{x+3}}={\frac {1+1}{1+3}}={\frac {2}{4}}=\mathbf {\frac {1}{2}} }$
20. ${\displaystyle \lim _{x\to 1}{\frac {5x}{x^{2}+2x-3}}}$
解答：当${\displaystyle x}$趋近于${\displaystyle 1}$时，分子趋近于${\displaystyle 5}$，分母趋近于${\displaystyle 0}$，但从左侧趋近时极限为${\displaystyle -\infty }$，从右侧趋近时极限为${\displaystyle +\infty }$，故极限不存在

## 无穷极限

1. ${\displaystyle \lim _{x\to \infty }{\frac {-x+\pi }{x^{2}+3x+2}}}$
解答：分母比分子高阶，故极限为${\displaystyle \mathbf {0} }$
2. ${\displaystyle \lim _{x\to -\infty }{\frac {x^{2}+2x+1}{3x^{2}+1}}}$
解答：分子与分母同阶，故极限为最高次项系数之比，即${\displaystyle \mathbf {\frac {1}{3}} }$
3. ${\displaystyle \lim _{x\to -\infty }{\frac {3x^{2}+x}{2x^{2}-15}}}$
解答：分子与分母同阶，故极限为最高次项系数之比，即${\displaystyle \mathbf {\frac {3}{2}} }$
4. ${\displaystyle \lim _{x\to -\infty }3x^{2}-2x+1}$
解答：极限为${\displaystyle \mathbf {+\infty } }$
5. ${\displaystyle \lim _{x\to \infty }{\frac {2x^{2}-32}{x^{3}-64}}}$
解答：分母比分子高阶，故极限为${\displaystyle \mathbf {0} }$
6. ${\displaystyle \lim _{x\to \infty }6}$
解答：极限为${\displaystyle \mathbf {6} }$
7. ${\displaystyle \lim _{x\to \infty }{\frac {3x^{2}+4x}{x^{4}+2}}}$
解答：分母比分子高阶，故极限为${\displaystyle \mathbf {0} }$
8. ${\displaystyle \lim _{x\to -\infty }{\frac {2x+3x^{2}+1}{2x^{2}+3}}}$
解答：分子与分母同阶，故极限为最高次项系数之比，即${\displaystyle \mathbf {\frac {3}{2}} }$
9. ${\displaystyle \lim _{x\to -\infty }{\frac {x^{3}-3x^{2}+1}{3x^{2}+x+5}}}$
解答：分子比分母高阶，故极限为${\displaystyle \mathbf {-\infty } }$
10. ${\displaystyle \lim _{x\to \infty }{\frac {x^{2}+2}{x^{3}-2}}}$
解答：分母比分子高阶，故极限为${\displaystyle \mathbf {0} }$

## 分段函数极限

1. ${\displaystyle f(x)={\begin{cases}(x-2)^{2}&{\mbox{, }}x<2\\x-3&{\mbox{, }}x\geq 2.\end{cases}}}$
1. ${\displaystyle \lim _{x\to 2^{-}}f(x)}$
解答：${\displaystyle (2-2)^{2}=\mathbf {0} }$
2. ${\displaystyle \lim _{x\to 2^{+}}f(x)}$
解答：${\displaystyle 2-3=\mathbf {-1} }$
3. ${\displaystyle \lim _{x\to 2}f(x)}$
解答：左右两侧极限不相等，故极限不存在
2. ${\displaystyle g(x)={\begin{cases}-2x+1&{\mbox{, }}x\leq 0\\x+1&{\mbox{, }}0
1. ${\displaystyle \lim _{x\to 4^{+}}g(x)}$
解答：${\displaystyle 4^{2}+2=16+2=\mathbf {18} }$
2. ${\displaystyle \lim _{x\to 4^{-}}g(x)}$
解答：${\displaystyle 4+1=\mathbf {5} }$
3. ${\displaystyle \lim _{x\to 0^{+}}g(x)}$
解答：${\displaystyle 0+1=\mathbf {1} }$
4. ${\displaystyle \lim _{x\to 0^{-}}g(x)}$
解答：${\displaystyle -2(0)+1=\mathbf {1} }$
5. ${\displaystyle \lim _{x\to 0}g(x)}$
解答：左右两侧极限相等，故极限为${\displaystyle \mathbf {1} }$
6. ${\displaystyle \lim _{x\to 1}g(x)}$
解答：${\displaystyle 1+1=\mathbf {2} }$
3. ${\displaystyle h(x)={\begin{cases}2x-3&{\mbox{, }}x<2\\8&{\mbox{, }}x=2\\-x+3&{\mbox{, }}x>2.\end{cases}}}$
1. ${\displaystyle \lim _{x\to 0}h(x)}$
解答：${\displaystyle 2(0)-3=\mathbf {-3} }$
2. ${\displaystyle \lim _{x\to 2^{-}}h(x)}$
解答：${\displaystyle 2(2)-3=4-3=\mathbf {1} }$
3. ${\displaystyle \lim _{x\to 2^{+}}h(x)}$
解答：${\displaystyle -(2)+3=\mathbf {1} }$
4. ${\displaystyle \lim _{x\to 2}h(x)}$
解答：左右两侧极限相等，故极限为${\displaystyle \mathbf {1} }$