Problem 26 - Precalculus: Mathematics for Calculus

Question

Finding Limits Evaluate the limit if it exists. $$\lim _{h \rightarrow 0} \frac{(2+h)^{3}-8}{h}$$

Step-by-Step Solution

Answer

The limit is 12.

Step 1: Substitute the Limit Variable

Let's break down the problem by first substituting the function given by \((2+h)^3 - 8\) into the limit expression. We want to evaluate \(\lim_{h \to 0} \frac{(2+h)^3 - 8}{h}\).

Step 2: Expand the Cubic Term

Expand \((2+h)^3\) using the binomial theorem:\[(2+h)^3 = 2^3 + 3 \cdot 2^2 \cdot h + 3 \cdot 2 \cdot h^2 + h^3 = 8 + 12h + 6h^2 + h^3\].

Step 3: Simplify the Expression

Replace \((2+h)^3\) in the limit expression with its expanded form:\[\lim_{h \to 0} \frac{8 + 12h + 6h^2 + h^3 - 8}{h}\].This simplifies to:\[\lim_{h \to 0} \frac{12h + 6h^2 + h^3}{h}\].

Step 4: Factor Out the Common Term

Simplify the expression by canceling the \(h\) in the numerator and denominator:\[\lim_{h \to 0} (12 + 6h + h^2)\].

Step 5: Evaluate the Limit

Now that the expression is simplified, we can directly evaluate the limit as \(h\) approaches 0:\[12 + 6 \cdot 0 + 0^2 = 12\].

Key Concepts

Polynomial Expansion

Polynomial expansion is a method that helps break down expressions into a simplified version by distributing powers across terms. In our exercise example, we aim to evaluate the limit of \((2+h)^3\), therefore expanding this expression is our crucial first step. We use the pattern: \((a+b)^n\), where \(n\) is the exponent. Here, the expression is expanded using algebraic strategies, allowing us to express it as a polynomial of several terms.
When expanding \((2+h)^3\):

• Start by identifying your base, which is "2" and your variable, "h".
• Here, our expansion unfolds as: \(2^3 + 3 \cdot 2^2 \cdot h + 3 \cdot 2 \cdot h^2 + h^3\).

The process allows us to reason about both value and structure as we further simplify or factor out terms. This maximizes comprehension of how additional variables mix with constant powers.

Binomial Theorem

The binomial theorem is fundamental for expanding expressions raised to a power, particularly when dealing with two terms. The theorem states: \[(a+b)^n = \sum_{k=0}^{n}\binom{n}{k}a^{n-k}b^k\]where \(\binom{n}{k}\) represents the binomial coefficients.
In terms of our solved example, we utilized it to expand the cubed binomial \((2+h)^3\). The specific coefficients in the expression come from the binomial coefficients, or 'Pascal's Triangle'. These coefficients control how terms like \(12h\), \(6h^2\), and \(h^3\) are arranged.

• Each term is formed by multiplying the coefficients from the binomial expansion formula with appropriate powers of "2" and "h".
• The constant and the variable parts blend to expand the expression, forming a polynomial.

Acknowledging the role of binomial theorem is vital to understanding how each term contributes to the expanded polynomial, simplifying further mathematical processes.

Evaluating Limits

Evaluating limits is a foundational concept in calculus, used to find the value a function approaches as the input approaches a specific point. In this problem, we evaluate the limit: \[\lim_{h \to 0} \frac{(2+h)^3 - 8}{h}\].
Initially, the direct substitution is not feasible due to an indeterminate form "0/0". To resolve this:

• First, expand \((2+h)^3\) as mentioned before.
• Replace it back in the limit equation, which shows potential issues with direct substitution get resolved through algebraic simplification.

After algebraic simplification, the limit function becomes straightforward, allowing us to substitute \(h=0\) directly, yielding the answer. Understanding this process is key in calculus for managing limits that initially seem complex.

Algebraic Simplification

Algebraic simplification aids in making a complex expression more manageable. In our example, after expanding the polynomial, the target is simplifying \[\lim_{h \to 0} \frac{12h + 6h^2 + h^3}{h}\].
Here's how it works:

• Notice common factors in terms that are factors of "h".
• Factor "h" out from each term in the numerator, making it \(h(12 + 6h + h^2)\).
• Cancel the common factor "h" from the numerator and the denominator.

This results in a much simpler expression: \(12 + 6h + h^2\).
Upon reaching the simplified form where the troublesome zero-denominator issue is bypassed, you can calculate a direct limit of \(h \to 0\). This approach is crucial when tackling problems involving limits, as it translates what's initially unmanageable into a form that can be easily analyzed.