import numpy as np
import sympy as sym
import matplotlib.pyplot as plt
from IPython.display import display,Math

display(Math('\\lim_{x \\rightarrow a} f(x)'))
print('Example 1')
display(Math('\\quad \\lim_{x \\rightarrow 4} \\frac{x^2}{2}=\\frac{4^2}{2}=8'))
print('Example 2')
display(Math('\\quad f(x)= \\frac{x^2}{x-2}'))
display(Math('\\quad \\lim_{x \\rightarrow 2} f(x)=\\frac{0}{0} \\text??'))
display(Math('\\quad \\lim_{x \\rightarrow 2^+} f(x)=+\\infty'))
display(Math('\\quad \\lim_{x \\rightarrow 2^-} f(x)=-\\infty'))

$\displaystyle \lim_{x \rightarrow a} f(x)$

Example 1

$\displaystyle \quad \lim_{x \rightarrow 4} \frac{x^2}{2}=\frac{4^2}{2}=8$

Example 2

$\displaystyle \quad f(x)= \frac{x^2}{x-2}$

$\displaystyle \quad \lim_{x \rightarrow 2} f(x)=\frac{0}{0} \text??$

$\displaystyle \quad \lim_{x \rightarrow 2^+} f(x)=+\infty$

$\displaystyle \quad \lim_{x \rightarrow 2^-} f(x)=-\infty$

x = sym.symbols('x')

fx = x**3

lim_pnt = 1.5
lim = sym.limit(fx,x,lim_pnt)

print(lim)
print(1.5**3)

display(Math('\\lim_{x\\to %g} %s = %g' %(lim_pnt,sym.latex(fx),lim)))

3.37500000000000
3.375

$\displaystyle \lim_{x\to 1.5} x^{3} = 3.375$

fxx = sym.lambdify(x,fx)
print(fxx(3))
xx = np.linspace(-5,5,200)

plt.plot(xx,fxx(xx))
plt.plot(lim_pnt,lim,'ro')
plt.show()

27

fx = (x**2)/(x-2)
fxx = sym.lambdify(x,fx)
xx = np.linspace(1,3,102)

lim_pnt = 2
lim = sym.limit(fx,x,lim_pnt,dir='+')
print(lim)
lim = sym.limit(fx,x,lim_pnt,dir='-')
print(lim)

plt.plot(xx,fxx(xx))
display(Math('\\lim_{x\\to %g^-} %s = %g' %(lim_pnt,sym.latex(fx),lim)))

oo
-oo

$\displaystyle \lim_{x\to 2^-} \frac{x^{2}}{x - 2} = -inf$