Problem 23 - Calculus Early Transcendentals

Question

Assume \(\lim _{x \rightarrow 1} f(x)=8, \lim _{x \rightarrow 1} g(x)=3,\) and \(\lim _{x \rightarrow 1} h(x)=2 .\) Compute the following limits and state the limit laws used to justify your computations. \(\lim _{x \rightarrow 1}(h(x))^{5}\)

Step-by-Step Solution

Answer

Answer: The limit of \((h(x))^5\) as \(x\) approaches 1 is 32.

Step 1: Recall the limit laws for powers of functions

To compute the limit of \((h(x))^5\) as \(x \rightarrow 1\), we can use the limit law for powers of functions, which states that if \(\lim_{x \rightarrow a}f(x) = L\) then \(\lim_{x \rightarrow a}(f(x))^n = L^n\), where n is a positive integer.

Step 2: Determine the limit of h(x) as x approaches 1

We are given that \(\lim_{x \rightarrow 1} h(x) = 2\).

Step 3: Apply the limit law for powers of functions

Since we know that the limit of h(x) as x approaches 1 is 2, we can apply the limit law for powers of functions to find the limit of \((h(x))^5\) as \(x \rightarrow 1\):$$\lim_{x \rightarrow 1}(h(x))^5 = (2)^5$$

Step 4: Compute the result

Calculate the limit of \((h(x))^5\) as \(x \rightarrow 1\):$$ (2)^5 = 32 $$ Thus, \(\lim_{x \rightarrow 1}(h(x))^5 = 32\). We used the limit law for powers of functions to justify our computation.

Key Concepts

Calculus

Calculus is a branch of mathematics focused on the study of change and motion, using processes like differentiation and integration to understand how things evolve over time or in response to various factors. In the realm of limits, calculus comes into play when evaluating how functions behave as they approach specific points. For example, in the provided exercise, we seek to determine the behavior of \(h(x)^5\) as \(x\) approaches 1. The calculus concept of limits allows us to predict the value of \(h(x)\) without explicitly calculating it at \(x=1\), which might be useful if the function is not defined at that point or has a complex expression.

Understanding calculus is crucial for a variety of scientific and engineering fields, where modeling and interpreting dynamic systems are necessary. It serves as a foundational tool in physics, economics, biology, and more, providing a logical framework for grasping the notions of instantaneous velocity, growth rates, and other time-sensitive phenomena.

Limits and Continuity

Limits are a fundamental concept in calculus dealing with the behavior of a function as its argument approaches some point. The focus is not on the function's actual value at that point, but on the value it approaches as it gets infinitely close. For instance, the limit of \(h(x)\) as \(x\) approaches 1 is 2, written mathematically as \(\lim_{x \rightarrow 1} h(x) = 2\). This notation means that as \(x\) gets closer and closer to 1, the values of \(h(x)\) approach 2.

Role of Continuity

Continuity, on the other hand, addresses whether a function is unbroken or uninterrupted when graphed. A function is continuous at a point if the limit of the function as it approaches that point equals the function's value at that point. In the exercise, continuity is implicitly suggested, since knowing the limits of \(f(x)\), \(g(x)\), and \(h(x)\) as \(x\) approaches 1, allows the use of limit laws to find the limits of these functions raised to a power, assuming the functions are continuous at that point.

Power of Functions Rule

The power of functions rule in limits is an essential law that helps simplify complex calculations. This limit law states that if you have a function \( f(x) \) that approaches a limit \( L \) as \( x \) approaches \( a \), then the function raised to any power \( n \) will approach \( L^n \) as \( x \) approaches \( a \). This is mathematically expressed as: \(\lim_{x \rightarrow a}(f(x))^n = L^n\). This useful property applies to the exercise at hand, where \(\lim_{x \rightarrow 1} h(x) = 2\), so \(\lim_{x \rightarrow 1}(h(x))^5 = 2^5\) simplifies to 32. It's by utilizing the power of functions rule that we are able to solve the problem directly without engaging in more complex evaluations.

Applicability of the Rule

This rule is not only useful in theoretical exercises but also in practical applications where we need to predict the outcome of processes that are affected by varying phenomena over time. For instance, in physics, it can be used to model the dissipating energy of a wave over distance, where we might consider the wave amplitude as a function of distance that is raised to a power.