物理でよく使われる近似式はどこまで成り立つか

import numpy as np
import matplotlib.pyplot as plt
import japanize_matplotlib
import pandas as pd

#フォントサイズを20に
plt.rcParams["font.size"] = 16
#メモリ線を内側に
plt.rcParams['ytick.direction'] = 'in'
#メモリ線を内側に
plt.rcParams['xtick.direction'] = 'in'
#上部に目盛り線を描く
plt.rcParams["xtick.top"] = True 
#右部に目盛り線を描く
plt.rcParams["ytick.right"] = True

color_list = ["black", "#E69F00", "#56B4E9", "#009E73", "#F0E442",
           "#0072B2", "#D55E00", "#CC79A7", "#999999"]

def fill_nan_with_neighbors(arr):
    """
    NaNを前後の値で補完する。
    最初や最後がNaNの場合も直近の有効な値で埋める。
    """
    arr = np.array(arr, dtype=np.float64)  # 確実にfloat型（NaN扱える）にする
    s = pd.Series(arr)
    filled = s.fillna(method='ffill').fillna(method='bfill')
    return filled.to_numpy()

def compare_function(x,y1,y2,y1_label,y2_label):
    plt.figure(figsize = (18,12))

    #1個めのplot
    plt.subplot(2,2,1)
    plt.plot(x, y1, label = y1_label, color=color_list[1])
    plt.plot(x, y2, label = y2_label, color=color_list[0])
    plt.ylabel("y")
    plt.xlabel("x")
    plt.legend()
    plt.grid()
    y_ma = np.max([np.max(y1), np.max(y2)])
    plt.ylim(0,y_ma + y_ma * 0.1)
    plt.xlim(np.min(x),np.max(x))

    #2個めのplot
    plt.subplot(2,2,2)
    plt.plot(x,np.abs(y2-y1),color=color_list[2])
    plt.ylabel("近似式と元関数の差分の絶対値")
    plt.xlabel("x")
    plt.grid()
    plt.ylim(0)
    plt.xlim(np.min(x),np.max(x))

    #3個めのplot
    plt.subplot(2,2,3)
    y3 = np.abs(y2-y1)/np.abs(y1)
    y3 = fill_nan_with_neighbors(y3)
    plt.plot(x,y3*100,color=color_list[3])
    plt.ylabel("近似式と元関数の変化の割合(%)")
    plt.xlabel("x")
    plt.grid()
    plt.ylim(0)
    plt.xlim(np.min(x),np.max(x))
    plt.hlines(y=10,xmin=np.min(x),xmax=np.max(x),color="r",linestyles="--")
    plt.show()

    # calculate the difference array
    difference_array = np.absolute(0.1-y3)

    # find the index of minimum element from the array
    index = difference_array.argmin()
    print("どこまで近似(<1割未満)が成り立つか",x[index])

# tan(x) ~ x, |x|<<1
x = np.arange(0,0.6,0.01)
y1 = np.tan(x)

compare_function(x,y1,x,"y = tanΘ","y = x")

# sin(x) ~ x, |x|<<1
x = np.arange(0,0.9,0.01)
y1 = np.sin(x)

compare_function(x,y1,x,"y = sinΘ","y = x")

# cos(x) ~ 1, |x|<<1
x = np.arange(0,0.5,0.01)
y1 = np.cos(x)
y2 = np.array([1 for _ in range(len(x))])

compare_function(x,y1,y2,"y = cosΘ","y = x")

# (1+x)^n ~ 1 + nx, |x|<<1
# n = 4 
x = np.arange(0,0.2,0.01)
n = 4
y1 = (1+x)**n
y2 = 1 + n*x

compare_function(x,y1,y2,"y = (1+x)$^n$","y = 1+nx")

# (n+x)^1/2 = n^1/2 (1 + x/n)^1/2
# ~ n^1/2 (1 + x / (2n) )
# |x| << n, → |x|/n << 1
# n = 4 
x = np.arange(0,7,0.01)
n = 4
y1 = n**(1/2) * (1 + x/n) ** (1/2)
y2 = n**1/2 * (1 + x /(2*n))

compare_function(x,y1,y2,"y = (n+x)$^{1/2}$","y = n$^{1/2}$ (1 + x/(2n))")