Problem 15 - Precalculus: Mathematical for Calculus

Question

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: Expand the Expression

To evaluate the limit \( \lim_{h \rightarrow 0} \frac{(2+h)^3 - 8}{h} \), start by expanding \((2+h)^3\). Using the binomial theorem, \((2+h)^3 = 2^3 + 3 \cdot 2^2 \cdot h + 3 \cdot 2 \cdot h^2 + h^3 = 8 + 12h + 6h^2 + h^3 \). Thus, the expression becomes \(\frac{(8 + 12h + 6h^2 + h^3) - 8}{h}\).

Step 2: Simplify the Expression

Next, simplify the expression by cancelling out the '8' from the numerator. This gives \(\frac{12h + 6h^2 + h^3}{h}\).

Step 3: Factor Out h from the Numerator

Factor \( h \) out of the numerator. The expression \( \frac{12h + 6h^2 + h^3}{h} \) becomes \( \frac{h(12 + 6h + h^2)}{h} \).

Step 4: Cancel h and Simplify

Cancel \( h \) from the numerator and denominator. This simplifies the expression to \( 12 + 6h + h^2 \).

Step 5: Take the Limit as h Approaches 0

Finally, evaluate the limit \( \lim_{h \rightarrow 0} (12 + 6h + h^2) \). As \( h \to 0 \), both \( 6h \) and \( h^2 \) approach zero, so the limit is \( 12 \).

Key Concepts

Binomial Theorem

The binomial theorem is a powerful tool in algebra that allows us to expand expressions that are raised to a power. It breaks down the process of expanding binomials like \[(a + b)^n\] into a series of terms based on binomial coefficients. Within the context of our problem, we utilized the binomial theorem to expand \((2 + h)^3\) into its polynomial form. This step involves recognizing that for any expression in the form \((a + b)^3\), each term in the expansion comes from multiplying one component from each pair of factors in \(a + b\).To expand \((2 + h)^3\), we used:

• The cube of the first term: \(2^3 = 8\)
• Three times the product of the square of the first term and the second term: \(3 \cdot 2^2 \cdot h = 12h\)
• Three times the product of the first term and the square of the second term: \(3 \cdot 2 \cdot h^2 = 6h^2\)
• The cube of the second term: \(h^3\)

This polynomial expansion is crucial for simplifying and evaluating the limit in calculus exercises.

Limit Evaluation

Limit evaluation is a fundamental concept in calculus, often used to find the behavior of functions as they approach a certain point. In our problem, we are asked to evaluate\(\lim _{h \rightarrow 0}\frac{(2+h)^3 - 8}{h}\).The initial step involved expanding \((2+h)^3\)using the binomial theorem as a polynomial. After the expansion, simplifying the expression is crucial for further evaluation. By removing common factors, specifically canceling \(h\) from both the numerator and the denominator, we are left with a simpler expression:\(12 + 6h + h^2\).Evaluating the limit then involves letting \(h\)approach 0. As we substitute 0 for each \(h\), the terms \(6h\)and \(h^2\)reduce to 0, leaving us with the final limit of \(12\).This process shows how limits help us understand the behavior of functions near a specific point.

Polynomial Expansion

Polynomial expansion transforms expressions into a sum of terms raised to different powers. In our specific problem, we started with the binomial \((2 + h)^3\).Using polynomial expansion techniques, particularly employing the binomial theorem, allowed us to break it down into a series of terms. The steps involve identifying coefficients and calculating powers of each term, resulting in:\[8 + 12h + 6h^2 + h^3\].Each term in this expansion represents a part of the overall expression:

• \(8\) represents the constant term.
• \(12h\) is the linear term involving \(h\).
• \(6h^2\) is the quadratic term.
• \(h^3\) is the cubic term.

This detailed breakdown through polynomial expansion not only aids in simplifying the given expression but also prepares it for further calculus operations, such as taking limits.