Problem 18 - Calculus: Early Transcendentals

Question

Suppose \(\lim _{x \rightarrow 2} f(x)=\lim _{x \rightarrow 2} h(x)=5 .\) Find \(\lim _{x \rightarrow 2} g(x),\) where \(f(x) \leq g(x) \leq h(x),\) for all \(x\).

Step-by-Step Solution

Answer

Answer: \(\lim_{x \rightarrow 2} g(x) = 5\)

Step 1: Recall the Squeeze Theorem

The Squeeze Theorem states that if \(f(x) \leq g(x) \leq h(x)\) for all \(x\) in an interval around a point \(a\), and \(\lim_{x \rightarrow a} f(x) = \lim_{x \rightarrow a} h(x) = L\), then \(\lim_{x \rightarrow a} g(x) = L\) as well.

Step 2: Apply the Squeeze Theorem

We are given that \(\lim_{x \rightarrow 2} f(x) = \lim_{x \rightarrow 2} h(x) = 5\). Also, it is given that \(f(x) \leq g(x) \leq h(x)\) for all \(x\). Since the conditions of the Squeeze Theorem are satisfied, we can conclude that \(\lim_{x \rightarrow 2} g(x) = 5\).

Step 3: State the result

The limit of function \(g(x)\) as \(x\) approaches 2 is 5, or \(\lim_{x \rightarrow 2} g(x) = 5\).

Key Concepts

Limits

In mathematics, limits are essential in understanding how functions behave at certain points or as they approach infinity. A limit describes the value that a function approaches as the input approaches a given point. Consider the limit notation \( \lim_{x \to a} f(x) = L \), which reads: "the limit of \( f(x) \), as \( x \) approaches \( a \), is \( L \)." This means that as \( x \) gets closer to \( a \), the function \( f(x) \) gets closer to \( L \).
Understanding limits is crucial in calculus, as they are foundational to defining derivatives and integrals. They help in analyzing functions when precise calculations directly at a point are complicated or impossible.

• One-sided limits: Considering approaching the point from left (\( x \to a^- \)) or right (\( x \to a^+ \)) allows deeper insights into behavior that might not be symmetric.
• Infinite limits: These describe situations where \( f(x) \) either becomes infinitely large or small as \( x \) approaches \( a \).

The Squeeze Theorem, featured in the original exercise, utilizes limits to "squeeze" a function's behavior between two others, effectively constraining its value to the same limit as its boundaries.

Calculus

Calculus is a significant branch of mathematics that deals with how things change. It provides tools like derivatives and integrals, which are built upon the concept of limits.
In calculus, we explore how functions behave under small changes – this is where the notion of differentiation comes in. Differentiation tells us how a function changes at any given point, often referred to as the rate of change or slope.

• Derivatives: Calculating a derivative involves finding the limit of the average rate of change of the function as the interval approaches zero.
• Integrals: While derivatives focus on rates of change, integrals are about summing up quantities. They accumulate values over intervals, useful in areas like determining area under curves.

The Squeeze Theorem is part of calculus, demonstrating the power of limits in understanding function behavior. It also shows how calculus techniques simplify solving real-world problems where direct computation isn't feasible.

Inequalities

Inequalities express relationships between values, providing a framework where one expression is not equal but rather less than or greater than another. In our exercise, inequalities are central to understanding the Squeeze Theorem application.
An inequality such as \( f(x) \leq g(x) \leq h(x) \) tells us that \( g(x) \) is always at least as large as \( f(x) \) and no larger than \( h(x) \). This forms a boundary in which \( g(x) \) must lie.

• Additive properties: If \( a \leq b \) and \( c \leq d \), then \( a + c \leq b + d \).
• Multiplicative properties: If both sides of an inequality are positive, multiplying each side by the same positive number keeps the inequality true.

These principles are pivotal in calculus, especially in squeezing functions between bounds or when setting up problem constraints.