Given f(x) = 3^3x − 1/12x

Question

Use a for loop (list comprehension) to evaluate the function at x = −0.1, −0.01, −0.001, −0.0001,

then at x = 0.1, 0.01, 0.001, 0.0001 to numerically estimate limx→0 f(x).

(b) Plot f on the interval [−1, 1] to graphically verify your estimate in part (a).

(c) Directly compute limx→0 f(x) and convert it to a decimal approximation to compare with

your estimate in part (a).

Answer 1

lim_x→0 [(3^3x-1) / (12x)] has the indeterminate form 0/0.

Applying L'Hopital's Rule, the given limit is equal to:

lim_x→0 [((ln3)3^3x(3) ) / 12] = 3ln3 / 12 = ln27 / 12 ≈ 0.27465

Answer 2

import matplotlib.pyplot as plt

import numpy as np

"""

a.) see get_values()

b.) see plot_values

c.)

Due to lack of parentheses is it unclear whether your last term is

-1/(12*x) OR (-1/12)*x

The former does not converge as x->0 as -1/(12*x)-> +infinity

as x->0 through negative values, and -1/(12*x)-> -infinity as

x->0 through postive values.

In the latter your function is continuous everywhere, and our

limit as x->0 is just f(0) = 3^(3*0)-(1/12)*0 = 1

(former part c is no value, latter part

I will be assuming the latter.

"""

def get_values(): # part

x_values_pos = [.1, .01, .001, .0001]

x_values_neg = [-.1, -.01, -.001, -.0001]

y = lambda x: 3 ** (3 * x) - (1 / 12)*x

# Lambda is a single line function.

y_values_pos = [y(x) for x in x_values_pos]

y_values_neg = [y(x) for x in x_values_neg]

#str([1,2,3]) = "[1,2,3]"

#in slicing, 0 is first -1 is last so this slice just takes off the brackets

print(f"Values for x={str(x_values_pos)[1:-2]} are y={str(y_values_pos)[1:-2]}")

print(f"Values for x={str(x_values_neg)[1:-2]} are y={str(y_values_neg)[1:-2]}")

def plot_values():

#problem probably wants you to do this by hand

#here's what you'd need to do to graph it in python

x = np.arange(-1.0, 1.0, 0.01)

# numpy supports "vectorized" ops - it just means it does these ops

# for every value in the vector

y = 3**(3*x)-(1/12)*x

fig, ax = plt.subplots()

ax.plot(x, y)

ax.grid()

fig.savefig("plot00.png")

plt.show()

get_values()

plot_values()