Problem 17 - Calculus Early Transcendentals

Question

Evaluate the limit, if it exists. $$\lim _{h \rightarrow 0} \frac{(4+h)^{2}-16}{h}$$

Step-by-Step Solution

Answer

The limit equals 8.

Step 1: Expand the Expression

Let's expand \((4 + h)^2\) first. Applying the formula for expanding binomials, we get: \((4 + h)^2 = 4^2 + 2 \times 4 \times h + h^2 = 16 + 8h + h^2\). Now, substitute back into the limit expression, which becomes \(\frac{16 + 8h + h^2 - 16}{h}\).

Step 2: Simplify the Numerator

Now, simplify the expression \(16 + 8h + h^2 - 16\) by cancelling out the constant **16** and **-16**, resulting in the new expression \(8h + h^2\). The limit then becomes: \(\frac{8h + h^2}{h}\).

Step 3: Factor and Reduce the Expression

Factor out an \(h\) from the numerator: \(h(8 + h)\). This allows us to cancel the \(h\) in the numerator and the denominator, leaving us with:\(8 + h\).

Step 4: Evaluate the Limit

Substitute \(h = 0\) into the simplified expression \(8 + h\), which gives us \(8 + 0 = 8\). Hence, the limit evaluates to \(8\).

Key Concepts

Limit Evaluation

Evaluating limits in calculus involves determining the behavior of a function as the variable approaches a specific value. For the given exercise, we're interested in analyzing the behavior of the function \[ \lim _{h \rightarrow 0} \frac{(4+h)^{2}-16}{h} \] as \(h\) approaches 0.
Limit evaluation often comes in handy when a function is not well-defined at a particular point, typically because it leads to an indeterminate form like \(\frac{0}{0}\). In these situations, we usually manipulate the expression algebraically to find a form that can be evaluated.
Here, the problems begin with substituting \(h = 0\), leading to \(\frac{0}{0}\). However, through simplifications and expansions, you can correctly evaluate the limit and find that this expression approaches 8. The key is to simplify expressions and remove indeterminacies.

Binomial Expansion

Binomial expansion is used to express powers of a binomial in expanded form. In this scenario, you need to expand \((4 + h)^2\) using the binomial theorem.
The binomial theorem states that for any positive integer \(n\), \((a + b)^n = a^n + \binom{n}{1}a^{n-1}b + \binom{n}{2}a^{n-2}b^2 + \ldots + b^n\).
Instead of applying the whole theorem, we use a simplified form for \(n = 2\): \[(4 + h)^2 = 4^2 + 2 \times 4 \times h + h^2 = 16 + 8h + h^2.\]
This expansion helps eliminate the indeterminate form in the limit problem by creating terms that can be factored and simplified further. Once expanded, you can isolate and cancel out terms that would initially lead to the problematic \(0\) in the denominator, making it easier for limit evaluation.

Simplifying Rational Expressions

Rational expressions are fractions where the numerator and denominator are both polynomials. The given function:\[\frac{(4+h)^2-16}{h}\] can be simplified by breaking down the numerator.

• First, expand \((4 + h)^2\), which we saw in the previous section gives us \(16 + 8h + h^2\).
• Subtract 16 to get \(8h + h^2\).
• Next, factor out a common \(h\): \(h(8 + h)\).
• Cancel \(h\) from both the numerator and the denominator.

Simplifying this expression means that we are left with \(8 + h\). This simplification is crucial as it allows us to bypass indeterminate forms and easily evaluate the limit.
Simplification strategies are instrumental in managing limits, especially those becoming indeterminate. They allow for easily computing limits and evaluating the function approaching the intended value.