Math Problem Statement

Complete the parts below to determine the apparent value of the following limit: lim (x+4)^3 - 64 / 4x as x approaches 0.

Solution

It looks like you're working on evaluating the apparent limit of the expression:

limx0(x+4)3644x\lim_{{x \to 0}} \frac{{(x + 4)^3 - 64}}{{4x}}

In part (a), you are calculating the values of (x+4)3644x\frac{{(x + 4)^3 - 64}}{{4x}} for different values of xx, and then for part (b), you'll use these results to estimate the limit as xx approaches 0.

To solve part (a), let me break it down and calculate the expression for each xx value:

f(x)=(x+4)3644xf(x) = \frac{{(x + 4)^3 - 64}}{{4x}}

I'll compute the values for x=1,0.5,0.05,0.001,0,0.001,0.05,0.5,1x = -1, -0.5, -0.05, -0.001, 0, 0.001, 0.05, 0.5, 1 and fill the blanks in your table.

Let me calculate these for you. numpy as np

Define the function

def f(x): return ((x + 4)**3 - 64) / (4 * x)

Values of x

x_values = np.array([-1, -0.5, -0.05, -0.001, 0.001, 0.05, 0.5, 1])

Calculate the corresponding y-values

y_values = [f(x) if x != 0 else None for x in x_values] # Avoid division by zero for x = 0 y_valuesHere are the calculated values for the expression (x+4)3644x\frac{{(x + 4)^3 - 64}}{{4x}} at the given xx values:

  • For x=1x = -1, the result is 9.2500
  • For x=0.5x = -0.5, the result is 10.5625
  • For x=0.05x = -0.05, the result is 11.8506
  • For x=0.001x = -0.001, the result is 11.9970
  • For x=0.001x = 0.001, the result is 12.0030
  • For x=0.05x = 0.05, the result is 12.1506
  • For x=0.5x = 0.5, the result is 13.5625
  • For x=1x = 1, the result is 15.2500

Now, for part (b), we observe the apparent value of the limit as xx approaches 0. From the values near x=0x = 0, we can estimate that the limit is approximately 12.0000.

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Math Problem Analysis

Mathematical Concepts

Limits
Algebra
Evaluation of Expressions

Formulas

lim(x→0) ((x+4)^3 - 64) / 4x

Theorems

Limit of a rational function as x approaches a value

Suitable Grade Level

Grades 11-12