# Part A:
#(a) Finding the limit of the functiom below
limit(sin(1/x-2)/(x-2),x=2)

#(b) Ploting the function below over the intervals of 1.5,2.5
plot(sin(1/x-2)/(x-2),x,1.5,2.5,ymin=-1,ymax=1)

#(c) Ploting the function below over the intervals of 1.95,2.05
plot(sin(1/x-2)/(x-2),x,1.95,2.05,ymin=-1,ymax=1)

#(d)Yes, the above function dose exist and my answer agrees with (a)

#Part B: Plotting (sin(2x)/x and (-cos(x)+3)
plot(sin(2*x)/x, x, -1.5, 1.5) + plot(-cos(x) + 3, x, -1.5, 1.5)

# sin(2x)/x <f(x)< -cos(x)+3 Finding limit of f(x) when x approching to 0
from sympy import limit, sin, cos, symbols

x = symbols('x')
f = (sin(2*x)/x) - cos(x) + 3

limit(f, x, 0)

#Part C:
#(a) Find the equation of the tangent line to the graph of f(x)=x^3+2x+1 at point (1,4)
x = var('x')
f(x) = x^3 + 2*x + 1
point = (1, 4)
derivative = f.derivative(x)
slope = derivative.subs(x=point[0])
equation = slope * (x - point[0]) + point[1]
equation

#(b)plottinf f(x)=x63+2x+1 with tangent line to the graph at point (1,4)
import numpy as np
import matplotlib.pyplot as plt

def f(x):
    return x**3 + 2*x + 1

def tangent_line(x):
    return 4 + 9*(x-1)

x = np.linspace(-5, 5, 100)
y = f(x)
tangent = tangent_line(x)

plt.plot(x, y, label="f(x) = x^3 + 2x + 1")
plt.plot(x, tangent, label="Tangent line at (1, 4)")
plt.scatter(1, 4, color='red', label="Point (1, 4)")
plt.legend()
plt.xlabel("x")
plt.ylabel("y")
plt.title("Plot of f(x) and Tangent Line")
plt.grid(True)
plt.show()

#Part D: (a) Plotting the graph of 1/x^2-4 over the interval [-3.5,3.5]

import numpy as np
import matplotlib.pyplot as plt

x = np.linspace(-3.5, 3.5, 100)  # Create an array of x values
y = 1/(x**2 - 4)  # Compute the corresponding y values

plt.figure(figsize=(6, 6))  # Set the figure size 6x6 inches
plt.plot(x, y, 'b-', label='f(x) = 1/(x^2 - 4)')
plt.axvline(x=-2, color='r', linestyle='--')  # Vertical asymptote x = -2
plt.axvline(x=2, color='r', linestyle='--')  # Vertical asymptote x = 2
plt.ylim(-15, 15)  # Set the y-axis limits
plt.xlabel('x')
plt.ylabel('f(x)')
plt.title('Graph of f(x) = 1/(x^2 - 4)')
plt.legend()
plt.grid(True)
plt.show()

#(b)Find fifth derivative of the function.
import sympy as sp

x = sp.symbols('x')
f = sp.diff(1 / (x**2 - 4), x, 5)

sp.pprint(f)