Problem 31 - Calculus: Early Transcendentals

Question

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

Step-by-Step Solution

Answer

The limit is \(3x^2\).

Step 1: Expand the Binomial

First, expand the expression \((x+h)^3\) using the binomial theorem:\[(x + h)^3 = x^3 + 3x^2h + 3xh^2 + h^3.\]

Step 2: Substitute Expanded Form into the Limit Expression

Substitute the expanded form back into the limit expression:\[\lim_{h \to 0} \frac{(x^3 + 3x^2h + 3xh^2 + h^3) - x^3}{h}.\]

Step 3: Simplify the Expression Inside the Limit

Simplify inside the fraction by canceling out the \(x^3\) terms:\[\frac{3x^2h + 3xh^2 + h^3}{h}.\]

Step 4: Factor Out h from the Numerator

Factor an \(h\) from the numerator:\[\frac{h(3x^2 + 3xh + h^2)}{h}.\]Now, cancel the \(h\) from the numerator and the denominator:

Step 5: Calculate the Limit

After canceling \(h\), evaluate the limit:\[\lim_{h \to 0} (3x^2 + 3xh + h^2) = 3x^2 + 3x(0) + (0)^2 = 3x^2.\]

Step 6: Conclusion

The limit exists and can be calculated as \(3x^2\).

Key Concepts

Binomial Theorem

The Binomial Theorem is a powerful tool in algebra for expanding expressions raised to any power. In the given problem, we need to expand \((x+h)^3\). According to the Binomial Theorem, \((x + h)^n\) can be expressed as the sum of terms involving binomial coefficients:

• The coefficient for each term is taken from Pascal's triangle, and its formula is \(\binom{n}{k}\), where \(n\) is the power, and \(k\) is the term's position.
• Here, in \((x+h)^3 = x^3 + 3x^2h + 3xh^2 + h^3\), we see the coefficients 1, 3, 3, and 1.

For our expression, using this theorem makes the expansion process easier and error-free.
It simplifies to give terms combining both \(x\) and \(h\), allowing us to further simplify our expression systematically.

Derivative

In calculus, the concept of derivatives represents the rate of change or the slope of the tangent line to a curve at a given point. The problem we tackled is keenly linked to derivatives as it embodies the difference quotient form,\(\lim_{h \to 0}\frac{f(x+h) - f(x)}{h}\),which is the foundational definition of a derivative. Here, our function \(f(x) = x^3\),and the expression we evaluated calculates the derivative of \(x^3\) at any point \(x\).

• This process confirms that \(\frac{d}{dx}(x^3) = 3x^2\).
• This derivative tells us how \(x^3\) changes at any specific value of \(x\).

This is incredibly essential for understanding not just motion and rates of change, but also optimizations in various fields like engineering and physics.

Simplifying Expressions

Simplification is the key to decoding complex mathematical problems into understandable terms. In our problem, after expanding \((x+h)^3\),we gathered a mix of terms and found that simplification helps in managing this complexity:

• We start by canceling out common terms,specifically \(x^3\)that appears in both the numerator terms.
• Then,we factor \(h\)out from the remaining terms\((3x^2h + 3xh^2 + h^3)\),simplifying the expression to\(h(3x^2 + 3xh + h^2)\).

Cancellation of \(h\) yields\(3x^2 + 3xh + h^2\),which simplifies our whole expression, making it easy to evaluate the limit.

Evaluating Limits

Evaluating limits is a fundamental skill in calculus that allows us to understand the behavior of functions as they approach specific points. In this instance, we had to evaluate\(\lim_{h \to 0}(3x^2 + 3xh + h^2)\).This process is straightforward once the expression is nicely simplified:

• Substitute the approaching value,here \(h = 0\),into the function to assess its behavior.
• The expression evaluated is then \(3x^2 + 3x(0) + (0)^2\),which simplifies directly to \(3x^2\).

This evaluation confirms that the limit exists and equals \(3x^2\),matching the derivative of the original function,demonstrating our correct application of these mathematical principles.