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Magazine Chapter 1: Problem 18
Evaluate the limit, if it exists. $$\lim _{h \rightarrow 0} \frac{(2+h)^{3}-8}{h}$$
Short Answer
Expert verified
The limit is 12.
Step by step solution
01
Recognize the Limit Form
The given problem is a limit problem that involves a rational expression. We are tasked to find \( \lim _{h \rightarrow 0} \frac{(2+h)^{3}-8}{h} \). This problem resembles the definition of the derivative of a function at a point. The expression \((2+h)^3\) needs to be expanded first to simplify the problem.
02
Expand the Cubic Expression
To simplify \((2+h)^3\), we use the binomial expansion formula. The expansion of \((a+b)^3\) is \(a^3 + 3a^2b + 3ab^2 + b^3\). Therefore, substituting \(a = 2\) and \(b = h\), we get:\((2+h)^3 = 2^3 + 3(2)^2h + 3(2)h^2 + h^3 = 8 + 12h + 6h^2 + h^3\).
03
Simplify the Expression
Substitute the expanded form of \((2+h)^3\) into the limit expression:\[ \lim _{h \rightarrow 0} \frac{8 + 12h + 6h^2 + h^3 - 8}{h} \]Cancel out the constant terms (8), leaving:\[ \lim _{h \rightarrow 0} \frac{12h + 6h^2 + h^3}{h} \].
04
Factor and Simplify Further
Factor out \( h \) from the numerator:\[ \lim _{h \rightarrow 0} \frac{h(12 + 6h + h^2)}{h} \].Cancel the \( h \) in the numerator and the denominator, resulting in:\[ \lim _{h \rightarrow 0} (12 + 6h + h^2) \].
05
Evaluate the Limit
Now substitute \( h = 0 \) into the simplified expression:\[ 12 + 6(0) + (0)^2 = 12 \].Thus, the limit evaluates to 12.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Binomial Expansion
Binomial Expansion is a powerful algebraic tool that helps expand expressions raised to a power. In our problem, we expanded \((2+h)^3\) using the binomial expansion formula. This formula says:
- \((a+b)^n = a^n + na^{n-1}b + \frac{n(n-1)}{2!}a^{n-2}b^2 + \cdots + b^n\)
- \(2^3 = 8\)
- \(3 \times 2^2 \times h = 12h\)
- \(3 \times 2 \times h^2 = 6h^2\)
- \(h^3\)
Rational Expressions
Rational Expressions are fractions in which both the numerator and the denominator are polynomials. In this particular exercise, the rational expression is \(\frac{(2+h)^{3}-8}{h}\). We need to simplify it to evaluate the limit.
- First, expand the cubic expression as we did with the binomial expansion.
- The expression becomes \(\frac{8 + 12h + 6h^2 + h^3 - 8}{h}\).
- Simplifying, the term \(-8\) cancels with \(8\), resulting in \(\frac{12h + 6h^2 + h^3}{h}\).
Limit Evaluation
Limit Evaluation is a foundational concept in calculus that helps find the value a function approaches as the variable gets arbitrarily close to a point. Our goal is to find the limit as \(h\) approaches zero.
- Once the numerator is simplified to \(12h + 6h^2 + h^3\), we factor out \(h\).
- This step results in \(\frac{h(12 + 6h + h^2)}{h}\).
- Cancel out the \(h\) in both the numerator and the denominator.
Derivative Definition
The Derivative Definition explains how to find the rate of change of a function. The solution to this problem resembles the derivative's definition at a point, where the expression is similar to:
- \(\lim_{h \to 0} \frac{f(x+h) - f(x)}{h}\)
- \((2+h)^3\) represents \(f(x+h)\)
- \(8\) is \(f(x)\) for \(x = 2\)
