Problem 6 - Contemporary Precalculus

Question

Use the following facts about the functions f, g, and h to find the required limit. $$\begin{aligned} &\lim _{x \rightarrow 4} f(x)=5 \quad \lim _{x \rightarrow 4} g(x)=0\\\ &\lim _{x \rightarrow 4} h(x)=-2 \end{aligned}$$ $$\lim _{x \rightarrow 4} h(x)^{2}$$

Step-by-Step Solution

Answer

Answer: The limit of h(x)^2 as x approaches 4 is 4.

Step 1: Identify the limit of h(x) as x approaches 4

According to the information given: $$\lim _{x \rightarrow 4} h(x)=-2$$

Step 2: Calculate the limit of h(x)^2 as x approaches 4

We know the limit of h(x) as x approaches 4 is -2, so to find the limit of h(x)^2 as x approaches 4, we square the limit of h(x): $$\lim _{x \rightarrow 4} h(x)^{2} = (-2)^{2}$$

Step 3: Compute the final result

After squaring -2, we get: $$\lim _{x \rightarrow 4} h(x)^{2} = 4$$ Therefore, the limit of h(x)^2 as x approaches 4 is 4.

Key Concepts

Function Limits

In calculus, the concept of a limit is foundational. A limit helps us understand the behavior of a function as it approaches a particular input. When we see \( \lim_{x \to a} f(x) \), it indicates the value that \( f(x) \) gets closer to as \( x \) approaches \( a \). This is crucial, especially when evaluating functions at points where they might not be directly calculated due to discontinuity or indeterminate forms.
To interpret \( \lim_{x \rightarrow 4} h(x) = -2 \), imagine getting closer and closer to \( x = 4 \). The values of \( h(x) \) move toward -2. Thus, by understanding limits, we see function behavior without directly substituting \( x = 4 \) if the function might misbehave there.

Limit Properties

Understanding limit properties is key to simplifying and accurately calculating complex limits. One of the fundamental properties is that limits distribute over addition, subtraction, multiplication, and division (if the limit of the denominator is not zero).
If we know \( \lim_{x \to 4} h(x) = -2 \), then \( \lim_{x \to 4} h(x)^2 = (-2)^2 \). This property, known as the power rule, allows us to handle more complicated expressions by individually squaring the limit. Key properties include:

• Sum/Difference Rule: \( \lim_{x \to a} [f(x) \pm g(x)] = \lim_{x \to a} f(x) \pm \lim_{x \to a} g(x) \)
• Product Rule: \( \lim_{x \to a} [f(x) \cdot g(x)] = \lim_{x \to a} f(x) \cdot \lim_{x \to a} g(x) \)
• Quotient Rule: \( \lim_{x \to a} \frac{f(x)}{g(x)} = \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)} \), given \( \lim_{x \to a} g(x) e 0 \)

These properties tremendously simplify finding limits of compositions or functions defined piecewise.

Mathematical Analysis

Mathematical analysis forms the rigorous underpinning for calculus. It includes the study of limits, continuity, and derivatives, rooted in an emphasis on precise definitions and methodology.
Consider the problem involving \( h(x) \). We start from a limit \( \lim_{x \to 4} h(x) = -2 \). The analysis ensures our understanding is based on a clear sequence of steps, rather than intuitive assumptions. By proving that \( \lim_{x \to 4} h(x)^2 = 4 \) using properties like continuity and functionality of \( h(x)^2 \), analysis ensures the result holds universally. This reliability is what makes mathematical analysis indispensable in theoretical and applied mathematics.