Problem 22 - Precalculus: Mathematical 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: Understanding the Problem

We need to find the limit as \(x\) approaches 0 for the expression \(\frac{(4+x)^{3}-64}{x}\). This expression is currently indeterminate at \(x = 0\) (\(\frac{0}{0}\) form), so we need to simplify to solve the problem.

Step 2: Recognizing a Cubic Expression

Observe that the expression \((4+x)^{3} - 64\) can be recognized as a difference of cubes: \((a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3\), where \(a = 4\) and \(b = x\). Here, \(64\) is \(4^3\), thus reinforcing the form.

Step 3: Expanding the Cubic Expression

Expand \((4+x)^3\) using the binomial theorem: \((4+x)^{3} = 4^3 + 3 \cdot 4^2 \cdot x + 3 \cdot 4 \cdot x^2 + x^3 = 64 + 48x + 12x^2 + x^3\).

Step 4: Simplifying the Expression

Substitute back into the difference \((4+x)^{3}-64 = 48x + 12x^2 + x^3\), resulting in the fraction \(\frac{48x + 12x^2 + x^3}{x}\). Simplify by cancelling \(x\) from each term in the numerator.

Step 5: Canceling and Further Simplification

Cancel \(x\) from the numerator: \(\frac{48x + 12x^2 + x^3}{x} = 48 + 12x + x^2\). Now, we can safely substitute \(x = 0\).

Step 6: Substituting the Limit Approach

Substitute \(x = 0\) into the simplified expression: \(48 + 12(0) + (0)^2 = 48\). Therefore, \(\lim _{x \rightarrow 0} \frac{(4+x)^{3}-64}{x} = 48\).

Step 7: Graphical Confirmation

Using a graphing calculator or software, plot the function \(f(x) = \frac{(4+x)^{3}-64}{x}\). Notice that as \(x\) approaches 0, the value of the function approaches 48, confirming our analytical solution.

Key Concepts

Difference of Cubes

The difference of cubes refers to a specific algebraic identity that helps in simplifying expressions involving cubes. Consider the expression \( a^3 - b^3\).This can be factored using the formula:\( (a - b)(a^2 + ab + b^2) \).For the problem in the original exercise, you need to recognize that \((4+x)^3 - 64\) is actually a difference of cubes because \(64 = 4^3\).
So, we can think of it as \((4+x)^3 - 4^3\).Using the difference of cubes formula, you can factor it into:

• \((4+x - 4)\)\(( (4+x)^2 + 4(4+x) + 4^2)\)

This simplifies to.

• \(x(16 + 16 + x^2 + 8x)\)

This clever trick reveals that the indeterminate form can actually be resolved by simplifying components, making the equation easier to tackle. It's an elegant algebraic strategy for unraveling complicated expressions without needing advanced techniques.

Binomial Theorem

The binomial theorem is a powerful tool in algebra that expands expressions raised to a power. It allows you to break down powers of binomials into polynomials. In the exercise, we use it to expand\((4+x)^3\).The binomial theorem formula states:\[(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\]Where:- \(a = 4\)- \(b = x\)- \(n = 3\)
Applying the formula, we expand as follows:

• First term: \(4^3 = 64\)
• Second term: 3\( \cdot 4^2 \cdot x = 48x\)
• Third term: 3 \(\cdot 4 \cdot x^2 = 12x^2\)
• Fourth term: \(x^3\)

The expansion becomes:\(64 + 48x + 12x^2 + x^3\).
This step is fundamental in breaking down complex polynomials and is especially useful when dealing with calculus limits or simplifying expressions for integration or differentiation.

Indeterminate Form

An indeterminate form arises in calculus when a limit calculation seems ambiguous, like \(\frac{0}{0}\). This occurs when both the numerator and denominator of a fraction approach zero as \(x\) approaches a specific value.
In the given problem, substituting \(x = 0\) directly into the expression \(\frac{(4+x)^3 - 64}{x}\) initially yields this form, because:

• Top part \( (4+0)^3 - 64 = 0\)
• Bottom part \(\ x = 0\)

This requires further manipulation to evaluate the limit.
When faced with an indeterminate form, simplify the expression by factoring, as shown using the difference of cubes or binomial expansion. This way, we simplify the expression and remove the indeterminate form, allowing us to substitute values with confidence. Here, simplifying gave us \(48 + 12x + x^2\), and substituting \(x = 0\), yielded a definitive answer, \48\. Indeterminate forms are common stumbling blocks, but they often simplify into determinate forms through algebraic manipulation.