Problem 24 - Precalculus Mathematics for Calculus

Question

Find the limit and use a graphing device to confirm your result graphically. $$\lim _{x \rightarrow 0} \frac{(4+x)^{3}-64}{x}$$

Step-by-Step Solution

Answer

The limit is 48.

Step 1: Identify the Limit Expression

We are tasked to find the limit as \( x \) approaches 0 of the function \( \frac{(4+x)^3 - 64}{x} \).

Step 2: Recognize Indeterminate Form

Substituting \( x = 0 \) directly into \( \frac{(4+x)^3 - 64}{x} \) gives \( \frac{(4)^3 - 64}{0} = \frac{64 - 64}{0} = \frac{0}{0} \), which is an indeterminate form. This means we need to simplify the expression.

Step 3: Simplify the Expression Using Algebra

Expand the numerator: \((4+x)^3 = 4^3 + 3 \cdot 4^2 x + 3 \cdot 4x^2 + x^3 = 64 + 48x + 12x^2 + x^3\). The expression becomes \( \frac{64 + 48x + 12x^2 + x^3 - 64}{x} \). Simplify to get \( \frac{48x + 12x^2 + x^3}{x} \).

Step 4: Further Simplification

Factor an \( x \) from the numerator: \( \frac{x(48 + 12x + x^2)}{x} \). Cancel the \( x \) from the numerator and denominator, giving \( 48 + 12x + x^2 \).

Step 5: Evaluate the Limit

Now evaluate the limit as \( x \rightarrow 0 \): \( 48 + 12(0) + (0)^2 = 48 \). Thus \( \lim_{x \rightarrow 0} \frac{(4+x)^{3}-64}{x} = 48 \).

Step 6: Confirm with Graphing

Use a graphing calculator or software to plot the function \( f(x) = \frac{(4+x)^3 - 64}{x} \). Observe the behavior as \( x \) approaches 0. The plot should approach the y-value of 48, confirming our analytical result.

Key Concepts

Indeterminate Forms

When dealing with limits in calculus, you’ll often encounter an expression that results in an indeterminate form like \( \frac{0}{0} \). This happens when direct substitution of the limit value into the function doesn't give a clear answer. This is a signal that the function hasn't been fully simplified yet and requires further manipulation.

In the given exercise, substituting \( x = 0 \) into \( \frac{(4+x)^3 - 64}{x} \) produces \( \frac{0}{0} \), which is indeterminate. We can't just say the limit is zero, infinite, or doesn’t exist; rather, we need more algebraic work to find a precise result. Understanding indeterminate forms is crucial as it guides us in pursuing algebraic simplification to properly evaluate the limit.

Algebraic Simplification

To resolve the indeterminate form \( \frac{0}{0} \), it’s necessary to simplify the expression further using algebraic methods. Here's how it's done in this exercise:

• First, we expand the cube in the numerator: \((4+x)^3 = 64 + 48x + 12x^2 + x^3\).
• This expansion allows the original expression to be rewritten as \( \frac{64 + 48x + 12x^2 + x^3 - 64}{x} \). Notice how the 64s cancel each other out, simplifying the problem.
• What's left is \( \frac{48x + 12x^2 + x^3}{x} \). The next step is to factor out the common \( x \) in the numerator.
• This leaves us with \( \frac{x(48 + 12x + x^2)}{x} \). We can now cancel the \( x \) in the numerator and denominator, resulting in the simplified expression \( 48 + 12x + x^2 \).

By simplifying algebraically, the limit can now be directly assessed without resulting in an indeterminate form providing a clearer path to evaluating it.

Graphical Confirmation of Limits

After performing the algebraic simplification and evaluating the limit to ensure its accuracy, using a graphing approach provides additional confirmation. Visualizing limits can be a powerful tool in understanding their behavior at specific points.

In this instance, plotting the function \( f(x) = \frac{(4+x)^3 - 64}{x} \) on a graphing tool demonstrates how the values behave as \( x \) approaches zero. The steep decline or ascent of the curve should stabilize around the limit value you calculated manually. For this exercise, you should notice that as \( x \) approaches 0, the y-values of the curve approach a horizontal line at \( y = 48 \).

This visualization effectively corroborates the analytical findings of the limit calculation, bolstering the confidence in the solution. When learning to understand limits profoundly, using both algebraic and graphical methods can provide comprehensive insight.