Question

Programming Use the programming capabilities of a graphing utility to write a program for approximating \(\lim _{x \rightarrow c} f(x)\). Assume the program will be applied only to functions whose limits exist as \(x\) approaches \(c\). Let \(y_{1}=f(x)\) and generate two lists whose entries form the ordered pairs \(\left(c \pm[0.1]^{n}, f\left(c \pm[0.1]^{n}\right)\right)\) for \(n=0,1,2,3,\) and 4

Short answer

The program returns two lists of ordered pairs. Each pair represents a value of \(x\) slightly above or slightly below \(c\), and the value of the function \(f(x)\) at that point. This provides an approximation of the limit of \(f(x)\) as \(x\) approaches \(c\).

Step-by-step solution

Importing the necessary python libraries

First of all, import the math and sympy libraries. The math library will be used to implement mathematical operations, and sympy provides functions for symbolic mathematics. The code would look as:\n\n import math \n from sympy import *

Defining the function

Define a function `f` which will return the value of `f(x)`. Let's say `f(x) = x^2`, so the code would look as: \n\n x = symbols('x') \n f = lambdify(x, x**2, 'numpy')

Create the limit approximation function

Now create a function that generates the two lists of ordered pairs for calculating the limit. The code might look like this: \n\n def limit_approximation(c): \n return [((c + 0.1**n, f(c + 0.1**n)), (c - 0.1**n, f(c - 0.1**n))) for n in range(0, 5)]

Run the function

Finally, call the function `limit_approximation` with the desired value of `c`. This will generate the two lists of ordered pairs. For c = 1, the code could be: \n\n limit_approximation(1)