Math Problem Statement
Solution
It looks like you're working on evaluating the apparent limit of the expression:
In part (a), you are calculating the values of for different values of , and then for part (b), you'll use these results to estimate the limit as approaches 0.
To solve part (a), let me break it down and calculate the expression for each value:
I'll compute the values for 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 at the given values:
- For , the result is 9.2500
- For , the result is 10.5625
- For , the result is 11.8506
- For , the result is 11.9970
- For , the result is 12.0030
- For , the result is 12.1506
- For , the result is 13.5625
- For , the result is 15.2500
Now, for part (b), we observe the apparent value of the limit as approaches 0. From the values near , 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