Problem 33 - Calculus

Question

Use algebra to evaluate the limit. $$\lim _{h \rightarrow 0} \frac{(3+h)^{2}-(3-h)^{2}}{2 h}$$

Step-by-Step Solution

Answer

The limit is 6.

Step 1: Expand Binomials

Expand each binomial in the expression using the formula \((a+b)^2 = a^2 + 2ab + b^2\). For \((3+h)^2\), substitute \(a=3\) and \(b=h\) to get \((3+h)^2 = 3^2 + 2(3)(h) + h^2 = 9 + 6h + h^2\). For \((3-h)^2\), substitute \(a=3\) and \(b=h\) to get \((3-h)^2 = 3^2 - 2(3)(h) + h^2 = 9 - 6h + h^2\).

Step 2: Substitute the Expansions

Substitute the expanded expressions back into the limit: \[\lim _{h \rightarrow 0} \frac{(9 + 6h + h^2) - (9 - 6h + h^2)}{2h}.\]

Step 3: Simplify the Expression

Simplify the expression inside the numerator: \[(9 + 6h + h^2) - (9 - 6h + h^2) = 9 + 6h + h^2 - 9 + 6h - h^2 = 12h.\]Now the expression becomes \[\lim _{h \rightarrow 0} \frac{12h}{2h}.\]

Step 4: Cancel Common Terms

Cancel the common \(h\) in the numerator and denominator:\[\lim _{h \rightarrow 0} \frac{12h}{2h} = \lim _{h \rightarrow 0} \frac{12}{2} = 6.\]

Step 5: Evaluate the Limit

Since there is no \(h\) left in the expression, the limit evaluates to its constant value:\[\lim _{h \rightarrow 0} = 6.\]

Key Concepts

Algebraic Manipulation

Algebraic manipulation is the process of rearranging and simplifying algebraic expressions to make problems easier to solve. It often involves techniques such as expanding, factorizing, and combining like terms. In our exercise, we begin with algebraic manipulation to tackle the limit evaluation problem.

• First, we recognize that the expression involves terms of the form \( (a+b)^2 \) and \( (a-b)^2 \).
• Expanding these binomials using the binomial theorem allows us to handle the expression in a more manageable way.
• Through careful manipulation, terms that are common or cancel out are isolated, simplifying the eventual calculation.

This is an essential skill in mathematics as it provides clarity and simplification of complex expressions, paving the way for smoother calculations.

Binomial Expansion

The binomial expansion is a formulaic way to expand terms raised to a power, such as \( (a+b)^2 \). In this formula, every term is broken down into simpler parts consisting of products of the base numbers and their powers.

• For \( (3+h)^2 \), apply \( (a+b)^2 = a^2 + 2ab + b^2 \) to get \( 9 + 6h + h^2 \).
• Similarly, \( (3-h)^2 \) becomes \( 9 - 6h + h^2 \).

This method is a quick way to evaluate quadratic expressions and is a foundational concept in algebra that greatly aids in simplifying limit problems by expanding polynomial-like terms to reveal cancellation opportunities.

Limit Problems

Limit problems are a key part of calculus, providing insights into the behavior of functions as they approach specific values. In this exercise, we're assessing the behavior of an expression as \( h \) approaches zero.

• The primary objective is to determine the value that the expression \( \frac{(3+h)^2-(3-h)^2}{2h} \) approaches as \( h \) tends towards zero.
• Recognizing terms and using techniques like simplification and cancellation are central to evaluating such limits.

By following these steps, we reduce complex ratios to a simpler form, whereby standard evaluation techniques can be confidently applied.

Simplifying Expressions

Simplifying expressions is the process of making an algebraic expression easier to understand or compute. In limit problems, simplification is crucial to avoid undefined forms.

• After binomial expansion, we simplify by combining like terms and canceling terms out. This is seen in how \( 9 + 6h + h^2 - (9 - 6h + h^2) \) simplifies to \( 12h \).
• The ultimate simplification leads to the cancellation of terms: \( \frac{12h}{2h} \), resulting in \( \frac{12}{2} = 6 \).

Understanding how to simplify expressions effectively allows for the solution of the limit as they are reduced to their most basic resolvable components. This concept underscores the importance of perceptive manipulation of mathematical expressions to solve problems efficiently.