Problem 34 - 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 Indeterminate Form

First, we substitute the value 0 for \(x\) directly into the expression \( \frac{(4+x)^3 - 64}{x} \). \((4+0)^3 - 64 = 64 - 64 = 0\), and the denominator is also zero. So we have \( \frac{0}{0} \), an indeterminate form, indicating we need further calculation.

Step 2: Expand the Cubic Expression

Expand and simplify \((4+x)^3\). This results in \((4+x)^3 = 4^3 + 3 \cdot 4^2 \cdot x + 3 \cdot 4 \cdot x^2 + x^3 = 64 + 48x + 12x^2 + x^3\).

Step 3: Substitute and Simplify

Substitute the expanded expression into the limit: \( \lim_{x \rightarrow 0} \frac{64 + 48x + 12x^2 + x^3 - 64}{x}\). Simplify the expression to \( \frac{48x + 12x^2 + x^3}{x} \).

Step 4: Cancel the Common Factor

Factor \(x\) out of the numerator: \( \frac{x(48 + 12x + x^2)}{x}\). Cancel the \(x\) from both the numerator and denominator, resulting in \(48 + 12x + x^2\).

Step 5: Evaluate the Limit

Now, substitute \(x = 0\) back into the simplified expression: \(48 + 12\times0 + 0^2 = 48\). Thus, the limit is \(48\).

Step 6: Confirm with a Graphing Device

Use a graphing calculator or software to graph \(y = \frac{(4+x)^3 - 64}{x}\) near \(x = 0\). Check that the graph approaches \(y = 48\) as \(x\) approaches 0, confirming our solution.

Key Concepts

Indeterminate Forms

When dealing with limits, one often encounters indeterminate forms. These are expressions where direct substitution leads to ambiguous results such as \( \frac{0}{0} \), \( \frac{\infty}{\infty} \), or \( 0 \times \infty \). These forms don't directly convey meaningful information about the function's behavior as the variable approaches a particular value.

In our exercise, substituting \( x = 0 \) into \( \lim _{x \rightarrow 0} \frac{(4+x)^{3}-64}{x} \) results in \( \frac{0}{0} \). This is a classic indeterminate form, signaling that we need further manipulation to reveal the function's limit. Recognizing these forms is crucial because they necessitate using specific techniques to resolve the ambiguity and find the actual limit.

• Identify when substitution leads to \( \frac{0}{0} \).
• Use algebraic manipulation to simplify the expression.
• Apply advanced techniques like L'Hôpital's Rule or factorization when applicable.

Understanding indeterminate forms is the first step in evaluating complex limits accurately.

Limit Evaluation Techniques

To handle indeterminate forms, mathematicians have developed various limit evaluation techniques. In our problem, after identifying an indeterminate form, we expanded the expression using algebraic manipulation, specifically cubic expansion. Once simplified, further algebraic steps led to finding the limit.

Common techniques include:

• Algebraic simplification: Break down complex expressions to reveal simpler terms that resolve the indeterminate form.
• Cancellation: Factor common terms, allowing simplification by cancellation.
• Substitution: After simplification, substitute back to find the limit.
• L'Hôpital's Rule: When dealing with \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \), taking derivatives of the numerator and denominator can often resolve the form directly.

In our exercise, after factoring out and canceling \( x \), we simplified the expression to \(48 + 12x + x^2\), which allowed us to substitute \( x = 0 \) easily, yielding the limit of 48.

Cubic Expansion

Cubic expansion is crucial when working with polynomials raised to a power, as it allows simplification of seemingly complex expressions. In our case, we expanded \( (4+x)^3 \) to make sense of the expression and eliminate the indeterminate form.

The cubic expansion follows the binomial theorem, expressed as:
\[ (a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3 \]
In our exercise, with \( a = 4 \) and \( b = x \), we expanded as:

• \(4^3 = 64\)
• \(3 \cdot 4^2 \cdot x = 48x\)
• \(3 \cdot 4 \cdot x^2 = 12x^2\)
• \(x^3 = x^3\)

Combining these results, \((4+x)^3 = 64 + 48x + 12x^2 + x^3\).

This expanded form is much easier to work with for limit evaluation, enabling us to isolate and cancel terms, moving toward finding the limit through substitution. Cubic expansion simplifies and clarifies the expressions, aiding in the resolution of complex problems effectively.