Problem 76 - Thomas Calculus

Question

Suppose that \(g(x) \leq f(x) \leq h(x)\) for all \(x \neq 2\) and suppose that \(\lim _{x \rightarrow 2} g(x)=\lim _{x \rightarrow 2} h(x)=-5\) Can we conclude anything about the values of \(f, g,\) and \(h\) at \(x=2 ?\) Could \(f(2)=0 ?\) Could lim \(_{x \rightarrow 2} f(x)=0 ?\) Give reasons for your answers.

Step-by-Step Solution

Answer

\( \lim_{x \rightarrow 2} f(x) = -5 \); \( f(2) \) can be 0, but the limit cannot.

Step 1: Understand the Problem

We are given three functions, \( g(x), f(x), \) and \( h(x) \) that satisfy \( g(x) \leq f(x) \leq h(x) \) for all \( x eq 2 \). Additionally, \( \lim_{x \rightarrow 2} g(x) = \lim_{x \rightarrow 2} h(x) = -5 \). We need to explore the possible values of \( f(x) \) at \( x = 2 \) and whether any conclusions can be drawn regarding \( \lim_{x \rightarrow 2} f(x) \) and \( f(2) = 0 \).

Step 2: Apply the Squeeze Theorem

The Squeeze Theorem states that if \( g(x) \leq f(x) \leq h(x) \) for all \( x eq 2 \) and \( \lim_{x \rightarrow 2} g(x) = \lim_{x \rightarrow 2} h(x) = L \), then \( \lim_{x \rightarrow 2} f(x) = L \). In this problem, \( L = -5 \), so it follows that \( \lim_{x \rightarrow 2} f(x) = -5 \).

Step 3: Explore f(2)

The problem asks about \( f(2) \). The value of \( f(2) \) can be anything because the conditions given are about the limit as \( x \) approaches \( 2 \), not the actual value of \( f \) at \( x = 2 \). Thus, \( f(2) = 0 \) is a possibility.

Step 4: Check if Limit Can Be Zero

We already determined that \( \lim_{x \rightarrow 2} f(x) = -5 \) using the Squeeze Theorem. As the limit approaches -5, \( \lim_{x \rightarrow 2} f(x) eq 0 \). Hence, the limit cannot be 0.

Key Concepts

Understanding Limits

Limits are a fundamental concept in calculus, representing the value that a function approaches as the input (or 'variable') gets closer to a specified point. In our exercise, we are interested in the limits as the input approaches a particular value, which is 2. When we say \( \lim_{x \rightarrow 2} g(x) = -5 \), it means as \( x \) gets closer and closer to 2, the function \( g(x) \) gets closer and closer to -5. The same applies to \( h(x) \). Essentially, the limit does not care about what happens at \( x=2 \) itself; it only concerns the behavior as \( x \) approaches 2. This is key to answering the question about bounds or potential values of functions at particular points.Here are some key points about limits:

• Limits help us analyze and predict function behavior without needing exact calculations.
• The limit of a function at a point gives us an approximate direction instead of the exact position.
• Understanding limits is crucial for tackling more advanced topics in calculus like derivatives and integrals.

Exploring Function Behavior with Squeeze Theorem

The concept of function behavior refers to understanding how a function acts as the input changes. In the context of the problem, we look at three functions: \( g(x), f(x), \) and \( h(x) \). According to the problem, these functions are ordered such that: \( g(x) \leq f(x) \leq h(x) \) for values of \( x eq 2 \). The Squeeze Theorem is vital here. It states that if a function is sandwiched between two other functions, both with the same limit at a point, then the sandwiched function must also reach the same limit at that point. In simpler terms, if \( g(x) \) and \( h(x) \) both converge to -5 as \( x \) approaches 2, then \( f(x) \) must do the same. This theorem helps us understand the behavior of \( f(x) \) despite not knowing its exact form.This logical conclusion relies on the continuous trend seen from the bounding functions, also demonstrating:

• The power of reasoning with inequalities in evaluating function limits.
• Why \( \lim_{x \rightarrow 2} f(x) = -5 \) is certain even if \( f(x) \) itself is not strictly defined at \( x=2 \).

Developing Mathematical Reasoning

Mathematical reasoning is about making logical deductions based on premises and known truths. In this problem, reasoning is applied to determine potential values of \( f(x) \) at \( x=2 \), and whether the limit \( \lim_{x \rightarrow 2} f(x) \) could be zero. First, the Squeeze Theorem was used, allowing us to conclude that \( \lim_{x \rightarrow 2} f(x) = -5 \). This means that, as \( x \) approaches 2, \( f(x) \) behaves much like \( g(x) \) and \( h(x) \). Therefore, the limit cannot be zero, as it would contradict the convergence established by the boundaries.For \( f(2) \), however, we have more flexibility. Since limits consider only the behavior around \( x=2 \), rather than any specific value at that point, \( f(2) \) could indeed be zero, or any other number:

• Mathematical reasoning shows us the importance of looking beyond exact values to the trends and conditions around them.
• It reinforces the concept that limits describe tendencies rather than fixed points, especially at undefined points.