Problem 24 - Calculus Early Transcendentals: Pearson New International Edition

Question

Let \(F\) and \(G\) be functions such that \(0 \leq F(x) \leq G(x)\) for all \(x\) near \(c\), except possibly at \(c\). Prove that if \(\lim _{x \rightarrow c} G(x)=0\), then \(\lim _{x \rightarrow c} F(x)=0\).

Step-by-Step Solution

Answer

By the Squeeze Theorem, \( \lim_{x \to c} F(x) = 0 \).

Step 1: Understanding the Problem

We are given two functions, \( F(x) \) and \( G(x) \), with the condition \( 0 \leq F(x) \leq G(x) \) for all \( x \) near \( c \), except possibly at \( c \) itself. The task is to show that if \( \lim_{x \to c} G(x) = 0 \), then \( \lim_{x \to c} F(x) = 0 \).

Step 2: Apply the Squeeze Theorem

The Squeeze Theorem states that if \( f(x) \leq g(x) \leq h(x) \) and \( \lim_{x \to c} f(x) = \lim_{x \to c} h(x) = L \), then \( \lim_{x \to c} g(x) = L \) as well. Here, choose \( f(x) = 0 \) and \( h(x) = G(x) \). We know \( \lim_{x \to c} f(x) = 0 \) and \( \lim_{x \to c} h(x) = 0 \). Since \( F(x) \) is squeezed between \( f(x) \) and \( h(x) \), the Squeeze Theorem applies.

Step 3: Conclude the Result

By the Squeeze Theorem, since \( 0 \leq F(x) \leq G(x) \) and both \( 0 \) and \( G(x) \) tend to \( 0 \) as \( x \to c \), we conclude \( \lim_{x \to c} F(x) = 0 \). Thus, the limit of \( F(x) \) as \( x \to c \) must also be \( 0 \).

Key Concepts

Understanding Limits in Calculus

Limits are a fundamental concept in calculus, describing the behavior of a function as its input approaches a particular point. They help us understand how functions behave at boundaries or points where they might not be well-defined.
The notation \( \lim_{x \to c} f(x) = L \) means that the function \( f(x) \) gets closer and closer to the value \( L \) as \( x \) approaches \( c \). Limits can exist even if the function itself is not defined at \( c \); what matters is the trend as \( x \) gets near \( c \).
Understanding limits allows us to tackle mysteries of functions that aren't otherwise easy to handle, particularly in continuous and discontinuous behaviors.

Exploring Function Inequalities

Function inequalities involve expressions where one function is greater than or less than another. In calculus, these inequalities help in comparing the growth or shrinkage rates of different functions.
For two functions \( F(x) \) and \( G(x) \), an inequality might be \( 0 \leq F(x) \leq G(x) \) for values near a certain point, \( c \). Such expressions tell us that no matter which \( x \) value near \( c \) we choose, \( F(x) \) is always less than or equal to \( G(x) \) and greater than or equal to zero.
These inequalities are crucial when using the Squeeze Theorem, as they establish the constraints necessary for applying this powerful calculus tool.

Understanding Calculus Proofs

Calculus proofs are logical arguments that demonstrate the truth of a calculus statement or theorem. They aim to clarify why certain mathematical facts hold true through logical reasoning and are essential in verifying concepts like the Squeeze Theorem.
In the exercise, the proof involves understanding that the functions \( f(x), g(x), \text{and } h(x) \) are related by inequalities and limits.
By setting \( f(x) = 0 \), \( g(x) = F(x) \), and \( h(x) = G(x) \), and knowing both \( f(x) \) and \( h(x) \) approach zero, it logically follows that \( g(x) \) must also approach zero. This sequence of logical steps forms the backbone of the calculus proof that is being established.