Problem 18 - Calculus: Early Transcendentals

Question

Evaluate the limit, if it exists. \( \displaystyle \lim_{h \to 0}\frac{(2 + h)^3 - 8}{h} \)

Step-by-Step Solution

Answer

The limit is 12.

Step 1: Identify the Expression

We are given the limit: \[\lim_{{h \to 0}} \frac{{(2 + h)^3 - 8}}{h}\]We need to evaluate this limit to see if it exists.

Step 2: Expand the Binomial Expression

The expression \((2 + h)^3\) needs to be expanded. Using the binomial theorem or expansion:\[(2 + h)^3 = 2^3 + 3 \cdot 2^2 \cdot h + 3 \cdot 2 \cdot h^2 + h^3 = 8 + 12h + 6h^2 + h^3\]

Step 3: Substitute and Simplify

Substitute the expanded binomial back into the limit expression:\[\lim_{{h \to 0}} \frac{{8 + 12h + 6h^2 + h^3 - 8}}{h}\]Simplify the numerator:\[8 + 12h + 6h^2 + h^3 - 8 = 12h + 6h^2 + h^3\]

Step 4: Factor Out the Common Term

Notice that the numerator \(12h + 6h^2 + h^3\) has \(h\) as a common factor. Factor \(h\) out:\[ 12h + 6h^2 + h^3 = h(12 + 6h + h^2)\]

Step 5: Cancel the Common Factor

Now, substitute the factored form back into the limit:\[\lim_{{h \to 0}} \frac{{h(12 + 6h + h^2)}}{h}\]Cancel the common \(h\) in the numerator and denominator:\[\lim_{{h \to 0}} (12 + 6h + h^2)\]

Step 6: Evaluate the Limit as h Approaches 0

Now that the expression is simplified, directly substitute \(h = 0\) to find the limit:\[12 + 6(0) + (0)^2 = 12\]

Key Concepts

Binomial Expansion

Binomial expansion is a powerful technique to expand expressions raised to a power, such as \((a + b)^n\). It helps in breaking down complex expressions into simpler terms. In our problem, we have \((2 + h)^3\). By using the binomial theorem, we can rewrite it as \(2^3 + 3 \cdot 2^2 \cdot h + 3 \cdot 2 \cdot h^2 + h^3\).

• First term: The cube of 2, which is 8.
• Second term: 3 times the square of 2 times \(h\), i.e., \(12h\).
• Third term: 3 times 2 times the square of \(h\), or \(6h^2\).
• Fourth term: The cube of \(h\), is \(h^3\).

It's essential to be familiar with binomial expansion, as it simplifies calculations and prepares us for solving limits effectively. It reveals the structure of the polynomial, making the cancellation of terms easier.

Factoring

Factoring is the method of breaking down expressions into multiples that can be multiplied together to get the original quantity. It is particularly helpful in simplifying expressions before evaluating limits.
In our example, after expanding \((2 + h)^3\), we simplify the numerator to \(12h + 6h^2 + h^3\). Here, factoring involves identifying common factors in terms and expressing the expression in a factored form: \(h(12 + 6h + h^2)\).
Recognizing the common term \(h\) across all terms lets us simplify the rational expression, making the calculation of the limit manageable. Factoring involves logical thinking to group terms and simplify expressions efficiently, revealing the essence of the expression one needs to analyze.

Substitution in Limits

Substitution in limits is a straightforward process in evaluating limits, especially after simplifying the expression. Once you have canceled common factors (as we've done in the limit \(\lim_{{h \to 0}} (12 + 6h + h^2)\)), the substitution method allows us to directly substitute the limit point into the simplified expression.
In our example, after canceling \(h\), substituting \(h = 0\) into \(12 + 6h + h^2\) gives us

• 12 from the constant term
• 0 from the \(6h\) term (since \(6 \times 0 = 0\))
• 0 from the \(h^2\) term (since \(0^2 = 0\))

Thus, the value of the limit is 12. The substitution method works well once the expression is properly simplified, permitting accurate completion of limit evaluation.

Simplifying Rational Expressions

Simplifying rational expressions is crucial in evaluating limits as it potentially helps eliminate terms that otherwise complicate the problem. It involves rewriting the expression in a form where it can be evaluated more directly.
In the limit \(\lim_{{h \to 0}} \frac{{(2 + h)^3 - 8}}{h}\), after expanding and simplifying, we found the expression \(\frac{{h(12 + 6h + h^2)}}{h}\). By simplifying, we cancel the common term \(h\) from both the numerator and denominator, leaving us with \(\lim_{{h \to 0}} (12 + 6h + h^2)\).
This simplification prepares us to use direct substitution to find the limit. Knowing how to simplify these expressions is key to evaluating limits effectively without indeterminate forms, leading to a clear path to the solution.