[Python知识库] 用python学习微积分（二）导数（下）- 极限和连续

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[Python知识库]用python学习微积分（二）导数（下）- 极限和连续

本文内容来自学习麻省理工学院公开课：单变量微积分-极限和连续-网易公开课

一、极限

1、抛出公式：

?????? $fx=x+1, x>=0$

?????? $fx = -x +2, x<0$

2、求极限的方法：

from sympy import *
x = symbols('x')
expr = x + 1
limit_expr = limit(expr, x, 0) 
limit_expr

3、给函数出个图：

import matplotlib.pyplot as plt
import numpy as np
from sympy import *

fig = plt.figure()
ax = fig.add_subplot(1, 1, 1)
ax.spines['left'].set_position('zero')
ax.spines['bottom'].set_position('zero')
ax.spines['right'].set_color('none')
ax.spines['top'].set_color('none')
ax.xaxis.set_ticks_position('bottom')
ax.yaxis.set_ticks_position('left')

x = np.linspace(0,np.pi,50)
y = x + 1
plt.plot(x,y, 'c', label=' fx = x + 1, x >= 0 ')

x = symbols('x')
expr = x + 1
limit_expr = limit(expr, x, 0) 
plt.scatter(0, limit_expr, c='r')

x = np.linspace(0,0-np.pi,50)
y = -x + 2
plt.plot(x,y, 'r', label=' fx = -x + 2, x < 0 ')

x = symbols('x')
expr = -x + 2
limit_expr = limit(expr, x, 0) 
plt.plot(0,limit_expr,lw=0, marker='o', fillstyle='none')

plt.legend(loc='upper right')

plt.show()

得出的结论是： $\lim_{x \rightarrow x_0}{f(x)}=f(x_0)$

二、连续函数和不连续函数

1、函数在点x0连续的条件是：

??? a、 $\lim_{x \rightarrow x_0}{f(x)}$ 存在，在x0点的左极限和右极限都存在，同时x0点的左极限等于x0点的右极限

??? b、fx在x0点有定义

??? c、 $\lim_{x \rightarrow x_0}{f(x)}=f(x_0)$ ? ( 需要注意的是当计算 $\lim_{x \rightarrow x_0}{f(x)}$ ，要避免直接计算 $f(x_0)$ )

2、不连续函数

??? a、跳跃间断，左右两个极限都存在但并不相等，例如上面图显的函数

???????? $fx=x+1, x>=0$

???????? $fx = -x +2, x<0$

??? b、可去间断函数，左右两个极限都存在并相等，例如

????????? fx = x / sin(x)

????????? fx = 1- cos(x) / x

import numpy as np
fig = plt.figure()
ax = fig.add_subplot(1, 1, 1)
ax.spines['left'].set_position('zero')
ax.spines['bottom'].set_position('zero')
ax.spines['right'].set_color('none')
ax.spines['top'].set_color('none')
ax.xaxis.set_ticks_position('bottom')
ax.yaxis.set_ticks_position('left')


# plot the functions x / sin(x)
x = np.linspace(-0.1,-2.5,100)
y = x / np.sin(x)
plt.plot(x,y, 'c', label='y= x / sin(x)')

x = np.linspace(0.1,2.5,100)
y = x / np.sin(x)
plt.plot(x,y, 'c')

x,y = symbols('x y')
expr = x/y
z = expr.subs(y,sin(x)).subs(x,0.1)
z1 = expr.subs(y,sin(x)).subs(x,0)
plt.plot(0,z,lw=0, marker='o', fillstyle='none',label=format(z1), color = 'b')

# plot the functions 1-cos(x)/x
x = np.linspace(-0.1,-2.5,100)
y = (1 - np.cos(x)) / x
plt.plot(x,y, 'r', label='y= (1 - cos(x))/x')

x = np.linspace(0.1,2.5,100)
y = (1 - np.cos(x)) / x
plt.plot(x,y, 'r')

x,y = symbols('x y')
expr = y/x
z = expr.subs(y,1 - cos(x)).subs(x,0.1)
z1 = expr.subs(y, 1- cos(x)).subs(x,0)
plt.plot(0,z,lw=0, marker='o', fillstyle='none', color='b')
plt.legend(loc='upper left')
# show the plot
plt.show()

c、无穷间断

????? $\lim_{x \rightarrow 0^+}{1/x} = \infty$ , $\lim_{x \rightarrow 0^-}{1/x} = -\infty$

import numpy as np
import matplotlib.pyplot as plt
fig = plt.figure()
ax = fig.add_subplot(1, 1, 1)
ax.spines['left'].set_position('zero')
ax.spines['bottom'].set_position('zero')
ax.spines['right'].set_color('none')
ax.spines['top'].set_color('none')
ax.xaxis.set_ticks_position('bottom')
ax.yaxis.set_ticks_position('left')
x = np.linspace(-np.pi,-0.1,50)
y = 1 / x
plt.plot(x,y, 'c', label=' fx = 1/x ')
x = np.linspace(0.1,np.pi,50)
y = 1 / x
plt.plot(x,y, 'c')
plt.legend(loc='upper right')
plt.show()

这个函数的导数：

from sympy import *
x = Symbol('x')
f = 1/x
derivative_f = f.diff(x)
derivative_f

导数的图像：

import numpy as np
import matplotlib.pyplot as plt
fig = plt.figure()
ax = fig.add_subplot(1, 1, 1)
ax.spines['left'].set_position('zero')
ax.spines['bottom'].set_position('zero')
ax.spines['right'].set_color('none')
ax.spines['top'].set_color('none')
ax.xaxis.set_ticks_position('bottom')
ax.yaxis.set_ticks_position('left')
x = np.linspace(-1,-0.1,50)
y = -1 / (x*x)
plt.plot(x,y, 'c', label=' fx = 1/x**2 ')
x = np.linspace(0.1,1,50)
y = -1 / (x*x)
plt.plot(x,y, 'c')
plt.legend(loc='center right')
plt.show()

d、其他（丑陋）间断

fx = sin(1/x) 当x趋近于零时，函数值上下反复震荡，但并没有在0点的左极限或右极限

import numpy as np
import sympy
import matplotlib.pyplot as plt
fig = plt.figure()
ax = fig.add_subplot(1, 1, 1)
ax.spines['left'].set_position('zero')
ax.spines['bottom'].set_position('zero')
ax.spines['right'].set_color('none')
ax.spines['top'].set_color('none')
ax.xaxis.set_ticks_position('bottom')
ax.yaxis.set_ticks_position('left')
x = np.linspace(-1,-0.01,50)
y = np.sin(1/x)
plt.plot(x,y, 'c', label=' fx = 1/x**2 ')
x = np.linspace(0.01,1,50)
y = np.sin(1/x)
plt.plot(x,y, 'c')
plt.legend(loc='center right')
plt.show()