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import numpy as np
import matplotlib.pyplot as plt
import math
%matplotlib inline
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x = np.linspace(0.1, 3)
f1x = [(i-1)/i for i in x]
f2x = [math.log(i) for i in x]
f3x = x-1
fig = plt.figure()
ax = fig.gca()
ax.plot(x,f1x, color='r', label='f1x')
ax.plot(x,f2x, color='g', label='f2x')
ax.plot(x,f3x, color='b', label='f3x')
ax.legend([r'$\frac{x-1}{x}$',r'$ln(x)$',r'$x-1$'])
ax.grid(True)
From the plot above, $\frac{x-1}{x} \leq ln(x) \leq x-1$ holds only for $ x \geq 1$ and not $ x > 0$
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