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

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

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

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

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

I'll compute the values for $x = -1, -0.5, -0.05, -0.001, 0, 0.001, 0.05, 0.5, 1$ 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 $\frac{{(x + 4)^3 - 64}}{{4x}}$ at the given $x$ values:

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

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

Would you like further explanation or assistance with any part of the problem?