Problem 17 - Calculus Early Transcendentals: Pearson New International Edition

Question

, find the indicated limit. In most cases, it will be wise to do some algebra first. $$ \lim _{h \rightarrow 0} \frac{(2+h)^{2}-4}{h} $$

Step-by-Step Solution

Answer

The limit is 4.

Step 1: Expand the Binomial Expression

Start by expanding the expression \((2+h)^2\). Apply the formula for the square of a binomial: \((a+b)^2 = a^2 + 2ab + b^2\). In this case, \((2+h)^2 = 4 + 4h + h^2\).

Step 2: Simplify the Numerator

Substitute the expanded form of \((2+h)^2\) into the numerator of the fraction: \(4 + 4h + h^2 - 4\). Simplify this to get \(4h + h^2\).

Step 3: Write the Simplified Limit Expression

Replace the original expression inside the limit with the simplified form: \(\lim_{h \to 0} \frac{4h + h^2}{h}\).

Step 4: Factor and Simplify

Factor \(h\) out of the numerator: \(\frac{h(4 + h)}{h}\). Cancel \(h\) from the numerator and the denominator to get \(4 + h\).

Step 5: Evaluate the Limit

Finally, evaluate the limit by substituting \(h = 0\) into the simplified expression: \(\lim_{h \to 0} (4 + h) = 4\).

Key Concepts

Binomial Expansion

Binomial expansion is a process of expanding expressions that are raised to a power. In this case, we have \((2+h)^2\), which follows the binomial theorem. The general formula for the square of a binomial is \((a+b)^2 = a^2 + 2ab + b^2\). This formula allows us to expand \((2+h)^2\) into \(4 + 4h + h^2\).

• \(a = 2\) and \(b = h\)
• \(a^2 = 2^2 = 4\)
• \(2ab = 2 \times 2 \times h = 4h\)
• \(b^2 = h^2\)

Understanding this step is crucial as it simplifies the expression into a form that is easier to work with. It lays the foundation for algebraic manipulation.

Simplifying Expressions

Simplifying expressions means reducing them to their simplest form. Once expanded, \((2+h)^2\) becomes \(4 + 4h + h^2\). Next, incorporate this into the original fraction \(\frac{(2+h)^2 - 4}{h}\).

• Substitute the expanded binomial: \(\frac{4 + 4h + h^2 - 4}{h}\)
• Simplify by subtracting \(4\) from \(4 + 4h + h^2\)
• This reduces the expression to \(4h + h^2\)

By simplifying, we remove extraneous terms, making it easier to perform further operations like factoring or evaluating limits.

Factoring

Factoring involves expressing an expression as the product of its factors. Here, we need to factor the simplified form \(4h + h^2\). Notice both terms share an \(h\). Pull out this common factor:

• Factor \(h\) from \(4h + h^2\): \(h(4 + h)\)
• This gives us the expression \(\frac{h(4 + h)}{h}\)

Factoring is useful for simplifying expressions because it often allows us to cancel terms. In this case, the \(h\) in the numerator and denominator cancels out, simplifying the expression to \(4 + h\). This is a vital step before plugging limits into the expression.

Evaluating Limits

Evaluating limits is a fundamental concept in calculus, used to find the value that a function approaches as the input approaches some value. With the expression simplified via factoring to \(4 + h\), now it's straightforward to evaluate the limit as \(h\) approaches \(0\).

• Original limit expression: \(\lim_{h \to 0} \frac{4h + h^2}{h}\)
• After simplifying, it becomes \(\lim_{h \to 0} (4 + h)\)
• Substitute \(h = 0\): it results in \(4\)

By evaluating the limit, we discover the value that the expression converges to as \(h\) nears zero. This evaluation indicates the continuity at the point and is essential for understanding the behavior of functions at specific points.